Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source

Nonlinear Analysis: Real World Applications 46 (2019) 545–582

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source✩ Guoqiang Ren, Bin Liu ∗ School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China

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Article history: Received 8 May 2018 Received in revised form 12 September 2018 Accepted 28 September 2018 Available online xxxx Keywords: Chemotaxis Boundedness Attraction–repulsion Global existence

abstract In this paper, we deal with the chemotaxis system with logistic source under homogeneous Neumann boundary conditions in a bounded convex domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by developing some Lp -estimate techniques, we show that the system possesses at least one global and bounded weak solution. Our results generalized and improved previous results, and partially results are new. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The classical model of population dynamics, originally formulated by Pierre-Francois Verhulst in 1838 [1], establishes that the self-limiting growth of a biological population size at a certain time, P (t), is described P through the so-called logistic equation dP dt = RP (1 − K ), where the constant R defines the growth rate and K is the carrying capacity of the species, which is also associated to the death rate of the same species. This pioneer formulation does not distinguish the portions of the space where the population is more or less distributed, neither the presence of further factors which induce the migration of the population from one zone to another. On the contrary, the term chemotaxis is exactly employed to explain the movement of cells occupying a space, which are stimulated by a chemical signal produced by a substance therein inhomogeneous distributed. Chemotaxis, the directed movement of cells or organisms in response to chemical stimuli, plays essential roles in various biological processes such as embryonic development, wound healing, and disease progression. This can lead to strictly oriented movement or to partially oriented and partially ✩ This work was partially supported by NNSF of China (Grant No. 11571126). ∗ Corresponding author. E-mail addresses: [email protected] (G. Ren), [email protected] (B. Liu).

https://doi.org/10.1016/j.nonrwa.2018.09.020 1468-1218/© 2018 Elsevier Ltd. All rights reserved.

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tumbling movement. The movement towards a higher concentration of the chemical substance is termed positive chemotaxis and the movement towards regions of lower chemical concentration is called negative chemotactical movement. Chemotaxis is an important means for cellular communication. Communication by chemical signals determines how cells arrange and organize themselves, like for instance in development or in living tissues. The origin of the chemotaxis model was introduced by Keller and Segel in [2], which describes aggregation of cellular slime mold toward a higher concentration of a chemical signal. For a comprehensive exposition of further examples illustrating the outstanding biological relevance of chemotaxis, we refer the reader to the survey of Hillen and Painter [3]. During the past four decades, the chemotaxis model has become one of the best study models in mathematical biology. Before going into detail, let us mention the following chemotaxis–(Navier)–Stokes model, which was initially proposed by Tuval et al. [4] ⎧ ut + V · ∇u = ∇ · (D(u)∇u − uS(x, u, v)∇φ(v)), x ∈ Ω , t > 0, ⎪ ⎪ ⎨ vt + V · ∇v = △v − ug(v), x ∈ Ω , t > 0, (1.1) V + κ(V · ∇V ) = η△V − ∇P + u∇ϕ, x ∈ Ω , t > 0, ⎪ t ⎪ ⎩ ∇ · V = 0, x ∈ Ω, This model describes the motion of oxygen-driven swimming cells in an incompressible fluid. The motion of the fluid is under the influence of gravitational force exerted from aggregating the cells into the fluid. Here, u and v are denoted as before, and V represents the velocity field of the fluid subject to an incompressible Navier–Stokes equation with pressure P and viscosity η and a gravitational force ∇ϕ. S(x, u, v) is the chemotactic sensitivity function which, according to the respective modeling background, may depend on u, v and x, g(v) denotes the consumption rate of the oxygen by the cell, and ϕ is a given potential function, and the constant κ is related to the strength of nonlinear fluid convection. The analysis of this model aroused a lot of interest recently, and many results were constructed [5–8]. Studies have shown that the reaction of one species to multiple stimuli is given by the motion of microglia in Alzheimer disease tethered to a glass side in a conflict situation involving β-amyloid (an attractant) and tumor necrosis factor α (a repellent). To model such biological processes, Luca et al. [9] proposed the following attraction–repulsion chemotaxis system ⎧ ut = ∇ · (D(u)∇u) − ∇ · (uχ(v)∇v) + ∇ · (uγ ξ(w)∇w), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ x ∈ Ω , t > 0, ⎪ ⎨ τ1 vt = △v − α1 v + β1 u, τ2 wt = △w − α2 w + β2 u, x ∈ Ω , t > 0, (1.2) ∂u ∂v ∂w ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ ∂ν u(x, 0) = u0 (x), τ1 v(x, 0) = v0 (x), τ2 w(x, 0) = w0 (x), x ∈ Ω , ∂ where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω and ∂ν denotes the derivative with respect to the outer normal of ∂Ω , u, v and w represent the concentration of microglia, the concentration of chemoattractant, and the concentration of chemorepellent, respectively. The functions χ and ξ are chemotaxis sensitivity functions, and the coefficients αi , βi (i = 1, 2) are positive. In this paper, we consider the following quasilinear chemotaxis system with logistic source, which is given by: ⎧ ut = ∇ · (D(u)∇u − Φ(u)∇φ(v) + Ψ (u)∇ψ(w)) + f (u), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ x ∈ Ω , t > 0, ⎪ ⎨ τ1 vt = △v − f1 (u, v) + f2 (u, v), τ2 wt = △w − α2 w + β2 u, x ∈ Ω , t > 0, (1.3) ∂u ∂v ∂w ⎪ ⎪ ⎪ = = = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ ∂ν u(x, 0) = u0 (x), τ1 v(x, 0) = v0 (x), τ2 w(x, 0) = w0 (x), x ∈ Ω , ∂ where Ω ⊂ RN (N ≥ 3) is a bounded convex domain with smooth boundary ∂Ω and ∂ν denotes the derivative with respect to the outer normal of ∂Ω , f1 (u, v) = ug(v), f2 (u, v) = −α1 v + β1 u, u(x, t), v(x, t) and w(x, t)

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represent the concentration of microglia, the concentration of chemoattractant, and the concentration of chemorepellent, respectively, and the coefficients αi , βi (i = 1, 2) are positive, τi ∈ {0, 1} (i = 1, 2). We assume that the diffusion coefficient D satisfies D(s) ∈ C 2 ([0, ∞)),

(1.4)

and some constants CD > 0 and m ∈ R exist such that D(s) ≥ CD sm

f or all

s ≥ 0.

(1.5)

We suppose that Φ(u), Ψ (u) ∈ C 2 ([0, ∞)) and satisfies 0 ≤ Φ(s) ≤ χsα , Φ(0) = 0, ξsβ ≤ Ψ (s) ≤ ζsβ , Ψ (0) = 0, f or all

s ≥ s0 ,

(1.6)

2+ω with χ, ξ, ζ > 0, α, β ∈ R, s0 > 1. We assume that the signal-dependent sensitivity φ, ψ ∈ Cloc ((0, ∞)) (0 < ω < 1) fulfills b a f or any v, w > 0 (1.7) 0 < φ′ ≤ k , 0 < ψ ′ ≤ k 1 v w 2 with some positive constant a, b and k1 , k2 ∈ R. In addition, the logistic source f (s) smooth on [0, ∞) satisfies f (0) ≥ 0 and f (s) ≤ ϱ − µsϵ , f or all s ≥ 0 (1.8)

with some positive constant ϱ, µ and ϵ. And g(v) ∈ C 1 ([0, ∞)) is nonnegative with g(0) = 0.

(1.9)

To this end, we assume that the initial data (u0 , v0 ) satisfy (u0 , v0 , w0 ) ∈ (W 1,l (Ω ))3

f or all l > N with u0 (x), v0 (x), w0 (x) > 0 in Ω .

(1.10)

In order to better understand problem (1.3), let us mention some previous contributions in this direction. When f1 ≡ 0, on one hand, the system without logistic source, Jin et al. [10] proved that system (1.2) admits a unique global classical solution under the assumptions that τ1 = τ2 = 0, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξf (u), where f (u) = um for m > 1, N ≥ 2 and τ1 = 1, τ2 = 0, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξf (u), where f (u) = um for m > 1, N = 2 or m ≥ 2, N ≥ 3. Furthermore, for both cases, there exists a constant C independent of t such that ∥u(·, t)∥ ≤ C. When τ1 = τ2 = 0 or τ1 = 1, τ2 = 0, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = uχ(v), Ψ (u) = ur ξ(w) under some assumptions, Wu et al. [11] showed that this system possesses a unique global classical solution that is uniformly bounded in high-dimensions; Li et al. [12] successfully extended to the porous media type, and proved that the system possesses at least one global weak solution and this solution is bounded. When τ1 = τ2 = 0, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = uχ(v), Ψ (u) = uξ(w), Tao and Wang [13] obtained the global boundedness of the solution in high-dimensions if repulsion prevails over attraction; if attraction prevails over repulsion, blowup of solutions was identified in R2 , and τ1 = τ2 = 1, the same paper deduced that the system is also globally well-posed if repulsion dominates over attraction when N = 2 and β1 = β2 . Recently, Jin [14] improved the results of [13], which imposed the smallness assumption on the initial mass proved that if repulsion dominates over attraction, the global existence of classical solutions in two dimensions and weak solutions in three dimensions with large initial data were obtained. Moreover, Liu and Tao [15] showed that the system admits global bounded classical solutions whenever the repulsion is dominated in two-dimensional smoothly bounded domains. When τ1 = τ2 = 1, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu, Jin and Liu [16] proved the existence of a global classical solution of the system (1.2) with large initial data in two or three dimensions whole space when repulsion cancels attraction. Relying on a new entropy-type

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inequality, Lin and Mu [17] obtain the global existence and boundedness of solutions to system (1.2) in two 1 , where k is a constant depending dimensions bounded domain if the initial data u0 satisfies ∥u0 ∥L1 (Ω) < kχα only on Ω , and the convergence to steady states is also given in two or three dimensions when τ1 = τ2 = 1, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu. On the other hand, the system with logistic source, Zhang and Li [18] showed all solutions of system (1.2) are bounded provided that N ≤ 2, µ > 0 or N ≤ 3, µ > NN−2 (χα1 − ξα2 ) when τ1 = τ2 = 0, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu and f (u) fulfills f (u) = ru − µu2 . Wang [19] proved that if τ1 = τ2 = 0, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu, f (u) fulfills f (u) ≤ a − buη , η > 1 and D(u) ≥ cum for all u ≥ 0 holds with some positive constant c and m ≥ 1, system (1.2) possesses a unique global bounded classical solution provided that b > b∗ , where { ∗

b =

N −2 N (χα1

0,

− ξα2 ),

if m ≤ 2 − if m > 2 −

2 N, 2 N.

When τ1 = τ2 = 1, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu and f (u) fulfills f (u) ≤ a − buθ , θ ≥ 1, Wang et al. [20] showed that system (1.2) possesses a globally bounded and classical solution if N ≤ 3 or { N + 2 N √N 2 + 6N + 17 − N 2 − 3N + 4 } θ > θN := min , with N ≥ 2. 4 4 As for N = 3, τ1 = τ2 = 1, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu and f (u) = u(1 − µuγ ) with γ ≥ 1 and µ > 0, it is shown in [12] that (1.2) admits a unique global bounded solution under the conditions αi ≥ 21 (i = 1, 2) and { 41 } 41 µ ≥ max ( χβ1 + 9ξβ2 )γ , (9χβ1 + ξβ2 )γ . 2 2 In addition, the convergence of the solution was discussed when γ ≥ 1 and u0 ̸= 0; however, the convergence rate was left as an open problem there. In the higher-dimensional setting N ≥ 3, Zheng et al. [21] proved that 2 system (1.2) admits a unique globally bounded classical solution provided that α1 = α2 and µ > χβ1θ+ξβ 0 for some constant θ0 > 0 when τ1 = τ2 = 1, D ≡ 1, φ(v) = v, ψ(w) = w, Φ(u) = χu, Ψ (u) = ξu and f (u) ≤ a − µu2 . The first two equations of (1.3) with w = 0 form the Keller–Segel chemotaxis system, whenf2 ≡ 0, the following chemotaxis model the motion of cells toward the higher concentration of oxygen that is consumed by the cells: ⎧ ut = ∇ · (D(u)∇u − Φ(u)∇φ(v)), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎨ vt = △v − ug(v), x ∈ Ω , t > 0, (1.11) ∂ ∂v ⎪ (D(u)∇u − S(x, u, v)∇φ(v)) = 0, = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ⎩ ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω, the initial–boundary value problems have been studied by many authors. In fact, when D ≡ 1, g(v) = v, Φ(u) = χu, χ > 0, ∇φ(v) = ∇v, Tao [22] proved that if ∥v0 ∥L∞ (Ω) is sufficiently small, then smooth bounded higher-dimensional (N ≥ 2) system (1.1) even admits a global classical solution which is bounded and smooth for t > 0; moreover, for large initial data, Tao and Winkler [23] showed that the problem (1.1) has a global weak solution which is eventually bounded and smooth under the assumptions that D ≡ 1, g(v) = v, Φ(u) = χu, χ > 0, ∇φ(v) = ∇v, and Ω ⊂ R3 is a bounded convex domain; Wang et al. [24] proved that system (1.1) possesses a unique global classical solution that is uniformly bounded, when Φ(u) = u, ∇φ(v) = χ(v)∇v, D(u) ≥ CD (u + 1)m−1 with m > 12 in the case N = 1 and m > 2 − N2 in the case N ≥ 2, the same author in [25] also showed that (1.1) has a unique global classical solution in the cases Φ(u) = u,

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g(v) = v, ∇φ(v) = ∇v, and m > 2 − N6+4 (N ≥ 3); Zheng [26] proved the system possesses global bounded 3N solutions if m > 2N +2 in the case (N ≥ 2) which extends the results of Wang et al. [25]. It is different from the model with w = 0 and f2 ≡ 0, when w = 0 and f1 model where the signal is produced not consumed by the cells: ⎧ ut = ∇ · (D(u)∇u − Φ(u)∇φ(v)) + f (u), x ∈ Ω , ⎪ ⎪ ⎪ ⎨ τ1 vt = △v − α1 v + β1 u, x ∈ Ω, ∂v ∂u ⎪ = = 0, x ∈ ∂Ω , ⎪ ⎪ ∂ν ∂ν ⎩ u(x, 0) = u0 (x), τ1 v(x, 0) = v0 (x), x ∈ Ω,

≡ 0, the following chemotaxis

t > 0, t > 0, t > 0,

(1.12)

which was used in mathematical biology to model aggregation of the slime mold Dictyostelium discoideum. The main issue of the investigation in the mathematical analysis was whether the system (1.9) allows for a chemotactic collapse, that is, if it possesses solutions u that blowup in infinite or finite time. In fact, when f (u) ≡ 0, N ≥ 2, Winkler in [27] shown that the corresponding initial–boundary value problems indeed possess some solutions which blowup in finite time provided that m < m0 := NN−2 , whereas if m > m0 := NN−2 , Nagai et al. [28] and Tao et al. [29] proved the solutions will be global for a large class of initial data. Moreover, Cie´slak [30] showed the solutions approach a spatially homogeneous steady state in the large time limit. However, despite of f (u) ̸= 0, Zheng et al. [31] show that the radially symmetric solutions blow up in finite time under some suitable conditions. When f (u) ̸= 0, τ1 = 1, D ≡ 1, φ(v) = v, Φ(u) = χu and f (u) = au − bu2 , Lankeit [32] proved system (1.9) with arbitrarily small initial data admits global weak solutions in N ≥ 1. Moreover, if N = 3, these solutions become classical after some time and provided that a is not too large. When f (u) ̸= 0, τ1 > 0, D ≡ 1, φ(v) = v, Φ(u) = χu and f (u) fulfills f (0) > 0, f (u) ≤ a − bu2 , Winkler [33] showed the system (1.9) possesses a unique bounded and global-in-time classical solution if b is big enough in N ≥ 1. Furthermore, even though Osaki [34] obtained the system (1.9) possesses global classical solutions for any b > 0, which remain in domain of N = 1,2, the same conclusion is not clear to occur for N ≥ 3. Throughout above analysis, compared with quasilinear chemotaxis system without logistic source, it is not so mature that the quasilinear chemotaxis system with logistic source. Inspired by the arguments in previous studies [13,14,18,19], we mainly investigated the global bounded classical and weak solutions to the system (1.3), our results in Theorem 2.1(1) generalize earlier results in [35, Theorem 1(i)] and [19, Theorem 1.2(ii)], Theorem 2.2 generalize earlier results in [10, Theorem 1.1(ii)], Corollary 2.1 is consistent with the results in [36, Theorem 1.3], and improved the results in [21, Theorem 1.1(ii)], the results of Theorem 2.3 are new. In this paper, we use symbols c and C as some generic positive constants. Sometimes, in order to distinguish them, we use symbols Ci and ci (i = 1, 2, . . .) which depend on m, CD , p, Ω and the initial ∫ ∫ data only. Moreover, for simplicity, u(x, t) is written as u, the integral Ω u(x)dx is written as Ω u(x). The rest of this paper is organized as follows. In Section 2, we summarize some basic definitions and some useful lemma in order to the main result. In Section 3, we give some fundamental estimates for the solution to the system (2.1) and proof of Theorems 2.1, 2.2. The proof of Theorem 2.3 is in Section 4. We close the paper by a concluding remark. 2. Preliminaries and main result Under the assumptions of D(u), the first equation of system (1.3) may be degenerate at u = 0. Therefore, system (1.3) does not allow for classical solvability in general as the well-known porous medium equations. We introduce the following definition of weak solutions.

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Definition 2.1. Let T ∈ (0, ∞). A triplet of nonnegative functions (u, v, w) defined in Ω × (0, T ) is called a weak solution of system (1.3) with f1 ≡ 0, if u ∈ L2 (0, T ; L2 (Ω )), v ∈ L2 (0, T ; W 1,2 (Ω )), w ∈ L2 (0, T ; W 1,2 (Ω )), D(u)∇u ∈ L2 (0, T ; L2 (Ω )), Φ(u)∇φ(v) ∈ L2 (0, T ; L2 (Ω )), Ψ (u)∇ψ(w) ∈ L2 (0, T ; L2 (Ω )); the integral equalities ∫ T∫ ∫ ∫ T∫ ∫ T∫ Φ(u)∇φ(v) · ∇ϕ D(u)∇u · ∇ϕ + uϕt − u0 ϕ(·, 0) = − − 0 Ω 0 Ω 0 Ω Ω ∫ T∫ ∫ T∫ f (u)ϕ Ψ (u)∇ψ(w) · ∇ϕ + + Ω

0

Ω

0

hold as well as ∫

T

∫

∫

−τ1

vϕt − τ1

and ∫

T

0

Ω

Ω

0

∫ wϕt − τ2

T

∫

∫

∫

∇v∇ϕ + α1

Ω

∫

−τ2

∫

v0 ϕ(·, 0) +

Ω

0

T

∫

∫

T

Ω

T

∫

Ω

uϕ Ω

0

∫

∫

∇w∇ϕ + α2 0

∫

vϕ = β1 Ω

0

∫

w0 ϕ(·, 0) +

T

T

∫

wϕ = β2 Ω

0

uϕ 0

Ω

hold for all ϕ ∈ C0∞ (Ω × [0, T )). If (u, v, w) is a weak solution of system (1.3) in Ω × (0, T ) for all T ∈ (0, ∞), then we call (u, v, w) a global weak solution. Definition 2.2. Let T ∈ (0, ∞). A pair of nonnegative functions (u, v) defined in Ω × (0, T ) is called a weak solution of system (1.3) with w ≡ 0 and f2 ≡ 0, if u ∈ L2 (0, T ; L2 (Ω )), v ∈ L2 (0, T ; W 1,2 (Ω )), D(u)∇u ∈ L2 (0, T ; L2 (Ω )), Φ(u)∇φ(v) ∈ L2 (0, T ; L2 (Ω )) and ug(v) ∈ L2 (0, T ; L2 (Ω )); the integral equalities ∫ T∫ ∫ ∫ − uζt − u0 ζ(·, 0) = − 0

Ω

Ω

and ∫

T

∫

−

0

Ω

∫ Ω

∫ Φ(u)∇φ(v) · ∇ζ Ω

0

∫

T

∫

v0 ζ(·, 0) = − Ω

T

∫ D(u)∇u · ∇ζ +

∫ vζt −

0

T

∫

T

∫

∇v · ∇ζ + 0

Ω

ug(v)ζ 0

Ω

hold for all ζ ∈ C0∞ (Ω × [0, T )). If (u, v) is a weak solution of system (1.3) in Ω × (0, T ) for all T ∈ (0, ∞), then we call (u, v) a global weak solution. Then, we give the main results of this paper. Theorem 2.1. Let f1 ≡ 0, τ1 = τ2 = 0, Ω ⊂ RN (N ≥ 3) be a bounded convex domain with smooth boundary. Assume that Φ(u), Ψ (u), φ(v), ψ(w) and f (u) satisfy (1.6)–(1.8). Suppose that D(u) satisfies (1.4) and (1.5). If one of the following conditions holds: (1) α − m < 1 − (1 − N2 )(1 − k1 ), β − m < 1 − (1 − N2 )(1 − k2 ) with N ≥ 3 when k1 < 1, k2 < 1, (2) α − m < 1, β − m < 1 when k1 ≥ 1, k2 ≥ 1, then for any choice of the initial data (u0 , v0 , w0 ) fulfilling (1.10), system (1.3) possesses at least one nonnegative global bounded weak solution (u, v, w).

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Remark 2.1. Theorem 2.1 shows that the system (1.3) admits a global bounded weak solution nothing to do with the sign of ξβ2 − χβ1 and the dampening intensity of logistic source. We now have to leave an open problem on existence of global bounded weak solution when k1 < 1, k2 > 1 or k1 > 1, k2 < 1. Remark 2.2. Theorem 2.1(1) generalizes earlier results in [35, Theorem 1(i)] with k1 = k2 = 0, and [19, Theorem 1.2(ii)] with α = β = 1, k1 = k2 = 0. Theorem 2.2. Let f1 ≡ 0, τ1 = 1, τ2 = 0, Ω ⊂ boundary. Assume that Φ(u) = χu, Ψ (u), φ(v), satisfies (1.4) and (1.5). If ϵ ≥ β + 1 and β + 2m fulfilling (1.10), system (1.3) possesses at least one

RN (N ≥ 3) be a bounded convex domain with smooth ψ(w) and f (u) satisfy (1.6)–(1.8). Suppose that D(u) ≥ 2, then for any choice of the initial data (u0 , v0 , w0 ) non-negative global bounded weak solution (u, v, w).

Remark 2.3. Theorem 2.2 generalizes earlier results in [10, Theorem 1.1(ii)] with m = 0 and f = 0. Corollary 2.1. Let f1 ≡ 0, τ1 = 0, w = 0 and ψ = 0, Ω ⊂ RN (N ≥ 3) be a bounded convex domain with smooth boundary. Assume that Φ(u), φ(v), and f (u) satisfy (1.6)–(1.8). Suppose that D(u) satisfies (1.4) and (1.5). If one of the following conditions holds: (1) α − m < N2 with N ≥ 3 when k1 < 1, (2) α − m < 1, when k1 ≥ 1, then for any choice of the initial data (u0 , v0 ) fulfilling (1.10), system (1.3) possesses at least one non-negative global bounded weak solution (u, v). Remark 2.4. When φ(v) = v, Corollary 2.1 is consistent with the results in [36, Theorem 1.3], and improve the results in [37, Theorem 1.1(ii)]. Corollary 2.2. Let f1 ≡ 0, τ1 = 1, w = 0, ψ = 0 and f = 0, Ω ⊂ RN (N ≥ 3) be a bounded convex domain with smooth boundary. Assume that Φ(u) and φ(v) satisfy (1.6), (1.7). Suppose that D(u) satisfies (1.4) and (1.5). If ϵ ≥ 2 and m ≥ 1, then for any choice of the initial data (u0 , v0 ) fulfilling (1.10), system (1.3) possesses at least one non-negative global bounded weak solution (u, v). Theorem 2.3. Let f2 ≡ 0, w ≡ 0, τ1 = 1, Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary. Assume that Φ(u) = χu, φ(v) ≡ v, g(u) and f (u) satisfy (1.6), (1.8), and (1.9). Suppose that D(u) satisfies (1.4), and (1.5). If ϵ ≥ 2 and m ≥ 1 − ϵ, then for any choice of the initial data (u0 , v0 ) fulfilling (1.10), system (1.3) possesses at least one non-negative global bounded weak solution (u, v). The proof of Theorems 2.1–2.3, we leave it in Sections 3 and 4. We will construct the approximation solutions to system (1.3) with f1 ≡ 0 and collect some basic facts for the approximated problem. Furthermore, we will prove the local existence of classical solutions to the approximated problem. We study the following non-degenerate system with boundary condition ⎧ ⎪ uεt = ∇ · (Dε (uε )∇uε − Φε (uε )∇φ(vε ) + Ψε (uε )∇ψ(wε )) + f (uε ), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ Ω , t > 0, ⎪ ⎨ τ1 vεt = △vε − α1 vε + β1 uε , τ2 wεt = △wε − α2 wε + β2 uε , ⎪ ∂uε ∂vε ∂wε ⎪ ⎪ ⎪ = 0, = 0, = 0, ⎪ ⎪ ∂ν ∂ν ⎪ ⎩ ∂ν uε (x, 0) = u0ε (x), vε (x, 0) = v0ε (x), wε (x, 0) = w0ε (x),

x ∈ Ω,

t > 0,

x ∈ ∂Ω , t > 0, x ∈ Ω.

(2.1)

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where ε ∈ (0, 1), Dε (s) = D(s+ε), Φε (s) = Φ(s+ε), and Ψε (s) = Ψ (s+ε) for all s ≥ 0. Since D ∈ C 2 ([0, ∞)) fulfills D(s) ≥ CD sm for all s > 0 with CD > 0, then we can estimate Dε (s) = D(s + ε) ≥ CD (s + ε)m ≥ CD sm

(2.2)

holds with the same values of CD and m, which, notably, are independent of ε. Furthermore, Dε (0) = D(ε) ≥ CD εm > 0. The initial data u0ε ∈ C α (Ω ) for some α ∈ (0, 1), v0ε , w0ε ∈ C 1 (Ω ), and (u0ε , v0ε , w0ε ) satisfy u0ε ≥ 0, v0ε ≥ 0, w0ε ≥ 0, { ∥u0ε ∥L∞ (Ω) ≤ ∥u0 ∥L∞ (Ω) + 1, ∥v0ε ∥W 1,∞ (Ω) ≤ ∥v0 ∥W 1,∞ (Ω) + 1, (2.3) ∥w0ε ∥W 1,∞ (Ω) ≤ ∥w0 ∥W 1,∞ (Ω) + 1. Under the framework of fixed point theorem, we will prove the local existence of classical solution to system (2.1) in the following lemma. Lemma 2.1. Let f1 ≡ 0, τ1 = τ2 = 0, Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary. Assume that Φε (uε ), Ψε (uε ), φ(vε ), ψ(wε ) and f (uε ) satisfy (1.6)–(1.8). Suppose that Dε (uε ) satisfies (1.4), and (1.5). Then, system (2.1) has a unique local-in-time classical solution (uε , vε , wε ) such that (uε , vε , wε ) ∈ C 0 ([0, Tmax ); W 1,l (Ω )) ∩ C 2,1 (Ω × [0, Tmax )) ∩ C 2,1 (Ω × [0, Tmax )), where Tmax denotes the maximal existence time and l > N . Moreover, if Tmax < +∞ , then ∥uε (·, t)∥L∞ (Ω) → ∞

as t → Tmax .

(2.4)

Proof . The proof is based on a combination of the fixed-point argument and the parabolic–elliptic regularity theory, which is standard. (1) Existence. Given T ∈ (0, 1) to be specified below, we set Π := C 0 (Ω × [0, T ]) and ST := {uε ∈ Π | ∥uε (·, t)∥L∞ (Ω) ≤ R f or all t ∈ [0, T ]}, where R := ∥u0ε ∥L∞ (Ω) + 1. It is clear that ST is a bounded closed convex subset of Π . For any given u ¯ε ∈ ST , a unique (vε , wε ) exists such that vε and wε solve { −△vε + α1 vε = β1 u ¯ε , x ∈ Ω , t ∈ (0, T ), (2.5) ∂vε = 0, x ∈ ∂Ω , t ∈ (0, T ), ∂ν and

−△wε + α2 wε = β2 u ¯ε , x ∈ Ω , t ∈ (0, T ), ∂wε = 0, x ∈ ∂Ω , t ∈ (0, T ), ∂ν respectively. Then, we can find a unique u by solving the following parabolic equation ⎧ uε )∇uε − Φε (uε )∇φ(vε ) + Ψε (uε )∇ψ(wε )) + f (uε ), x ∈ Ω , t ∈ (0, T ), ⎪ ⎨ uεt = ∇ · (Dε (¯ ∂uε = 0, x ∈ ∂Ω , t ∈ (0, T ), ⎪ ⎩ ∂ν uε (x, 0) = u0ε (x), x ∈ Ω. {

(2.6)

(2.7)

Thus, we can introduce a mapping Υ : u ¯ε (∈ ST ) ↦−→ uε by defining Υ (¯ uε ) = uε . We can show that for a sufficiently small T , Υ has a fixed point. In fact, by the elliptic regularity [38, Theorem 8.34], we can show that there is a unique solution vε (·, t) ∈ C 1+ς (Ω ) to (2.5) for each ς ∈ (0, 1). Similarly, there is a unique wε (·, t) ∈ C 1+ς (Ω ) to (2.6). Moreover, in view of the Sobolev embedding theorem and the Lp estimates of the elliptic equation, we have ∥∇vε ∥L∞ ((0,T );C ς (Ω)) ≤ C1 ∥vε ∥L∞ ((0,T );W 2,l (Ω)) ≤ C2 ∥¯ uε ∥L∞ ((0,T )×Ω)

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553

and ∥∇wε ∥L∞ ((0,T );C ς (Ω)) ≤ C1 ∥wε ∥L∞ ((0,T );W 2,l (Ω)) ≤ C2 ∥¯ uε ∥L∞ ((0,T )×Ω) with l > N and some constants C1 , C2 > 0 are determined by l. Since (−Φ(u)∇φ(v) + Ψ (u)∇ψ(w)) ∈ ¯ ) for each ς ∈ (0, 1) due to the Sobolev embedding theorem: (W 1,l (Ω )) ↪→ L∞ ((0, T ) × Ω ) and u0ε ∈ C ς (Ω ς ς ¯ ¯ × [0, T ]) and ∥uε ∥ ς, ς ≤ C3 C (Ω ), then from [39, Theorem V1.1], we can derive that uε ∈ C ς, 2 (Ω ¯ C 2 (Ω×[0,T ]) for some ς ∈ (0, 1) and C3 > 0, where C3 depends on min0≤s≤R D(s), ∥∇vε ∥L∞ ((0,T );C ς (Ω)) and ∥∇wε ∥L∞ ((0,T );C ς (Ω)) . Noticing that the latter two quantities can be controlled by ∥¯ uε (·, t)∥L∞ ≤ R, we ς see that C3 depends only on R and thus ∥uε (·, t) − u0ε ∥L∞ (Ω) ≤ C3 (R)t 2 . Then we have ς

∥uε (·, t)∥L∞ (Ω) ≤ ∥u0ε ∥L∞ (Ω) + ∥uε (·, t) − u0ε ∥L∞ (Ω) ≤ ∥u0ε ∥L∞ (Ω) + C3 (R)t 2 . −2

From this we deduce that if we fixed T < C3 ς (R), then we have ∥uε (·, t)∥L∞ (Ω) ≤ ∥u0ε ∥L∞ (Ω) + 1. This implies that uε ∈ ST . By a straightforward reasoning, we can further show that Υ is continuous and bounded. Then the Schauder fixed point theorem ensures that there exists a uε ∈ ST such that Υ (uε ) = uε . ς ¯ × [0, T ]), we also have vε (·, t) ∈ C 2+ς, 2ς (Ω ¯ ) and wε (·, t) ∈ C 2+ς, 2ς (Ω ¯ ) by the classical Since uε ∈ C ς, 2 (Ω regularity theory of elliptic equations. Then we can use the regularity theory for parabolic equations 1+ς ¯ × [δ, T ]) for all δ ∈ (0, T ), which implies that [39, Theorem V6.1] to deduce that uε ∈ C 2+ς, 2 (Ω 1+ς 1+ς 2+ς, 2 2+ς, 2 ¯ ¯ vε ∈ C (Ω × [δ, T ]) and wε ∈ C (Ω × [δ, T ]). The solution may be prolonged in the interval [0, Tmax ) with either Tmax = ∞ or Tmax < ∞, where the latter case entails that (2.4) holds. Moreover, since f (0) ≥ 0, the maximum principle ensures that uε , vε and wε are non-negative. (2) Uniqueness. We turn to prove the uniqueness of solutions of system (2.1). Let (u1ε , v1ε , w1ε ) and (u2ε , v2ε , w2ε ) are two classical solutions to system (2.1) in Ω × (0, T ) with the same initial data. Then v1ε − v2ε satisfies the equation 0 = △(v1ε − v2ε ) − α1 (v1ε − v2ε ) + β1 (u1ε − u2ε ).

(2.8)

Differentiating (2.8) with respect to t and taking v1ε − v2ε as a test function, we can obtain ∫ ∫ 1 d α1 d 2 2 |∇(v1ε − v2ε )| + |v1ε − v2ε | 2 dt 2 dt Ω ∫ Ω (u1ε − u2ε )t (v1ε − v2ε )

= β1

Ω ∫

(2.9) ∫

∇(A(u1ε ) − A(u2ε )) · ∇(v1ε − v2ε ) + β1 (Φε (u1ε )φ′ (v1ε )∇v1ε − Φε (u2ε ) Ω ∫ × φ′ (v2ε )∇v2ε ) · ∇(v1ε − v2ε ) − β1 (Ψε (u1ε )ψ ′ (w1ε )∇w1ε − Ψε (u2ε )ψ ′ (w2ε )∇w2ε ) Ω ∫ ∇(v1ε − v2ε ) + β1 (f (u1ε ) − f (u2ε ))(v1ε − v2ε )

= −β1

Ω

Ω

∫s for all t ∈ (0, T0 ) with T0 ∈ (0, T ), where A(s) := 0 Dε (ρ)dρ. We now estimate each term on the right-hand side of (2.9), for the first term, we use (2.8), mean value theorem and Young’s inequality to deduce ∫ −β1 ∇(A(u1ε ) − A(u2ε )) · ∇(v1ε − v2ε ) ∫ Ω = β1 (A(u1ε ) − A(u2ε )) · △(v1ε − v2ε ) (2.10) Ω∫ ∫ = α1 β1 (A(u1ε ) − A(u2ε ))(v1ε − v2ε ) − β12 (A(u1ε ) − A(u2ε ))(u1ε − u2ε ) Ω∫ ∫ Ω 2 = α1 β1 C4 (u1ε − u2ε )(v1ε − v2ε ) − β1 C4 (u1ε − u2ε )2 Ω ∫ Ω ∫ α12 C4 β12 C4 2 2 |v1ε − v2ε | − |u1ε − u2ε | ≤ 2 2 Ω Ω

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for some positive constants C4 ∈ [Dε (m1 ), Dε (m2 )], where m1 := min{∥u1ε ∥L∞ (Ω×(0,T0 )) , ∥u2ε ∥L∞ (Ω × (0, T0 ))} and m2 := max{∥u1ε ∥L∞ (Ω×(0,T0 )) , ∥u2ε ∥L∞ (Ω×(0,T0 )) }. For the second term, similar to the first term, we use mean value theorem, H¨ older’s inequality and Young’s inequality to obtain ∫ β1 (Φε (u1ε )φ′ (v1ε )∇v1ε − Φε (u2ε )φ′ (v2ε )∇v2ε ) · ∇(v1ε − v2ε ) ∫Ω = β1 (Φε (u1ε ) − Φε (u2ε ))φ′ (v1ε )∇v1ε + Φε (u2ε )(φ′ (v1ε ) − φ′ (v2ε ))∇v1ε (2.11) Ω

+ Φε (u2ε )φ′ (v2ε )∇(v1ε − v2ε ) ∫ ∫ ∫ α12 C4 2 2 2 |u1ε − u2ε | + C5 |∇(v1ε − v2ε )| + C6 |v1ε − v2ε | ≤ 8 Ω Ω Ω with some positive constants C5 and C6 . By a similar procedure, we have ∫ −β1 (Ψε (u1ε )ψ ′ (w1ε )∇w1ε − Ψε (u2ε )ψ ′ (w2ε )∇w2ε ) · ∇(v1ε − v2ε ) Ω ∫ ∫ ∫ α12 C4 2 2 2 |u1ε − u2ε | + C7 |∇(v1ε − v2ε )| + C8 |∇(w1ε − w2ε )| ≤ 8 Ω Ω Ω ∫ 2 + C9 |w1ε − w2ε |

(2.12)

Ω

for some positive constants C7 , C8 and C9 . Since u1ε and u2ε are bounded in Ω ×(0, T0 ) and f (uε ) is smooth, we obtain ∫ ∫ ∫ α2 C4 2 2 |u1ε − u2ε | + C10 |v1ε − v2ε | (2.13) β1 (f (u1ε ) − f (u2ε ))(v1ε − v2ε ) ≤ 1 8 Ω Ω Ω with some positive constants C10 . Combining the above inequalities, we conclude that ∫ ∫ 1 d α1 d 2 2 |∇(v1ε − v2ε )| + |v1ε − v2ε | 2 dt Ω 2 dt Ω ∫ ∫ β 2 C4 2 2 ≤( 1 + C6 + C10 ) |v1ε − v2ε | + (C5 + C7 ) |∇(v1ε − v2ε )| (2.14) 2 Ω Ω ∫ ∫ ∫ α2 C4 2 2 2 + C8 |∇(w1ε − w2ε )| + C9 |w1ε − w2ε | − 1 |u1ε − u2ε | . 8 Ω Ω Ω Taking a similar procedure, we can deduce that ∫ ∫ 1 d α2 d 2 2 |∇(w1ε − w2ε )| + |w1ε − w2ε | 2 dt Ω 2 dt Ω ∫ ∫ ∫ 2 2 2 ≤ C11 |v1ε − v2ε | + C11 |∇(v1ε − v2ε )| + C11 |∇(w1ε − w2ε )| Ω Ω Ω ∫ ∫ α22 C4 2 2 + C11 |w1ε − w2ε | + |u1ε − u2ε | , 8 Ω Ω

(2.15)

for some positive constant C11 . Adding (2.14) to (2.15), we obtain ∫ ( ) 1 d 2 2 2 2 |∇(v1ε − v2ε )| + |∇(w1ε − w2ε )| + α1 |v1ε − v2ε | + α2 |w1ε − w2ε | 2 dt Ω ∫ α2 C4 + α22 C4 2 + 1 |u1ε − u2ε | 8 Ω ∫ ∫ ∫ ) (∫ 2 2 2 2 ≤ C12 |v1ε − v2ε | + |∇(v1ε − v2ε )| + |∇(w1ε − w2ε )| + |w1ε − w2ε | Ω

Ω

Ω

Ω

for all t ∈ (0, T0 ) and with some positive C12 . Then by Gronwall’s inequality, we conclude that u1ε = u2ε , v1ε = v2ε and w1ε = w2ε in Ω × (0, T0 ). Since T0 ∈ (0, T ) is arbitrary, the uniqueness of solutions is obtained.

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555

Lemma 2.2. Let f1 ≡ 0, τ1 = 1, τ2 = 0, Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary. Assume that Φε (uε ) = χuε , Ψε (uε ), φ(vε ), ψ(wε ) and f (uε ) satisfy (1.6)–(1.8). Suppose that Dε (uε ) satisfies (1.4), and (1.5). Then, system (2.1) has a unique local-in-time classical solution (uε , vε , wε ) such that (uε , vε , wε ) ∈ C 0 ([0, Tmax ); W 1,l (Ω )) ∩ C 2,1 (Ω × [0, Tmax )) ∩ C 2,1 (Ω × [0, Tmax )), where Tmax denotes the maximal existence time and l > N . Moreover, if T < +∞ , then ∥uε (·, t)∥L∞ (Ω) + ∥vε (·, t)∥W 1,∞ (Ω) → ∞

as t → Tmax .

(2.16)

Proof . Similar to the parabolic–elliptic case, the local existence can be proved by the well-established methods involving standard parabolic regularity theory and an appropriate fixed point framework, while the uniqueness can be obtained by adapting the method of [33]. As the proof is quite standard, we will refrain from a detailed and only sketch the main steps. (1) Existence. Given T ∈ (0, 1) to be specified below, we set Π := C 0 (Ω × [0, T ]) and ST := {uε ∈ Π | ∥uε (·, t)∥L∞ (Ω) ≤ R f or all t ∈ [0, T ]}, where R := ∥u0ε ∥L∞ (Ω) + 1. It is clear that ST is a bounded closed convex subset of Π . For any given u ¯ε ∈ ST , a unique (vε , wε ) exists such that vε and wε solve { vεt = △vε − α1 vε + β1 u ¯ε , x ∈ Ω , t ∈ (0, T ), (2.17) ∂vε = 0, x ∈ ∂Ω , t ∈ (0, T ), ∂ν and

−△wε + α2 wε = β2 u ¯ε , x ∈ Ω , t ∈ (0, T ), ∂wε = 0, x ∈ ∂Ω , t ∈ (0, T ), ∂ν respectively. Then, we can find a unique u by solving the following parabolic equation ⎧ uε )∇uε − Φε (uε )∇φ(vε ) + Ψε (uε )∇ψ(wε )) + f (uε ), x ∈ Ω , t ∈ (0, T ), ⎪ ⎨ uεt = ∇ · (Dε (¯ ∂uε = 0, x ∈ ∂Ω , t ∈ (0, T ), ⎪ ⎩ ∂ν uε (x, 0) = u0ε (x), x ∈ Ω. {

(2.18)

(2.19)

Thus, we can introduce a mapping Υ : u ¯ε (∈ ST ) ↦−→ uε by defining Υ (¯ uε ) = uε . We can show that for a sufficiently small T , Υ has a fixed point. In fact, by the parabolic regularity [39], we can show that there is a unique solution vε (·, t) ∈ C 1+ς (Ω ) to (2.5) for each ς ∈ (0, 1). By the elliptic regularity [38], there is a unique wε (·, t) ∈ C 1+ς (Ω ) to (2.6). Moreover, in view of the Sobolev embedding theorem and the Lp estimates of the parabolic and elliptic equations, we have ∥∇vε ∥L∞ ((0,T );C ς (Ω)) ≤ C1 ∥vε ∥L∞ ((0,T );W 2,l (Ω)) ≤ C2 ∥¯ uε ∥L∞ ((0,T )×Ω) and ∥∇wε ∥L∞ ((0,T );C ς (Ω)) ≤ C1 ∥wε ∥L∞ ((0,T );W 2,l (Ω)) ≤ C2 ∥¯ uε ∥L∞ ((0,T )×Ω) with l > N and some constants C1 , C2 > 0 are determined by l. The following is similar to parabolic–elliptic case, so we omit it. (2) Uniqueness. We turn to prove the uniqueness of solutions of system (2.1). Let (u1ε , v1ε , w1ε ) and (u2ε , v2ε , w2ε ) are two classical solutions to system (2.1) in Ω × (0, T ) with the same initial data. Letting z1 = u1ε − u2ε , z2 = v1ε − v2ε , z3 = w1ε − w2ε , for t ∈ (0, T ), we obtain upon subtracting the respective

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equations in (2.1) and performing obvious testing procedures the identities ∫ ∫ ∫ ∫ ] α1 d [1 2 2 2 |∇z2 | + z = β1 z1 z2t z2t + dt 2 Ω 2 Ω 2 Ω Ω ∫ ∫ ∫ = −β1 ∇z1 · ∇z2 − β1 z1 z2 + β1 z12 Ω

Ω

(2.20)

Ω

and d dt

∫ Ω

∫

∫ ∇(A(u1ε ) − A(u2ε )) · ∇z1 + (Φε (u1ε )φ′ (v1ε )∇v1ε Ω ∫ Ω ′ − Φε (u2ε )φ (v2ε )∇v2ε ) · ∇z2 − (Ψε (u1ε )ψ ′ (w1ε )∇w1ε Ω ∫ ′ − Ψε (u2ε )ψ (w2ε )∇w2ε ) · ∇z2 + (f (u1ε ) − f (u2ε ))z2 .

2 z1t =−

(2.21)

Ω

∫s

for all t ∈ (0, T0 ) with T0 ∈ (0, T ), where A(s) := 0 Dε (ρ)dρ. Similar to the procedures of (2.10)–(2.13), we have ∫ ∫ ∫ ∫ ∫ ∫ d 2 2 2 2 z1t ≤ c1 |∇z1 | + c2 z12 + c2 |∇z1 | + c2 |∇z2 | + c2 z22 (2.22) dt Ω Ω∫ Ω∫ ∫ Ω ∫ Ω ∫ Ω 2 2 + c3 z12 + c3 |∇z1 | + c3 z32 + c3 |∇z3 | + c4 z12 . Ω

Ω

Ω

Ω

Ω

By the Young inequality, we have 1 4

∫

1 − z1 z2 ≤ 2 Ω

∫

∫ −

∇z1 · ∇z2 ≤ Ω

as well as

∫

∫

2

|∇z1 | + Ω

Ω

|∇z2 |

2

(2.23)

Ω

z12

1 + 2

∫

z22 .

(2.24)

Ω

Combining (2.15) with (2.20)–(2.24), we obtain that ∫ ∫ ∫ ∫ ∫ ) d( 2 2 z12 + z22 + |∇z2 | + z32 + |∇z3 | dt Ω Ω ∫ Ω ∫ Ω ∫ Ω ∫ (∫ ) 2 2 2 2 2 ≤C z1 + z2 + |∇z2 | + z3 + |∇z3 | Ω

Ω

Ω

Ω

Ω

for all t ∈ (0, T0 ) with certain positive C. Then by Gronwall’s inequality, we conclude that u1ε = u2ε , v1ε = v2ε and w1ε = w2ε in Ω × (0, T0 ). Since T0 ∈ (0, T ) is arbitrary, the uniqueness of solutions is obtained. Next, we construct the approximation solutions to system (1.3) with w ≡ 0, f2 ≡ 0, Φε (uε ) = χuε and collect some basic facts for the approximated problem. Furthermore, we will prove the local existence of classical solutions to the approximate problem. We study the following non-degenerate system with boundary condition ⎧ uεt = ∇ · (Dε (uε )∇uε − χuε ∇φ(vε )) + f (uε ), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎨ τ1 vεt = △vε − uε g(vε ), x ∈ Ω , t > 0, (2.25) ∂vε ∂uε ⎪ = 0, = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ uε (x, 0) = u0ε (x), vε (x, 0) = v0ε (x), x ∈ Ω. where ε ∈ (0, 1), Dε (s) = D(s + ε) and Φε (s) = Φ(s + ε) for all s ≥ 0. Since D ∈ C 2 ([0, ∞)) fulfills D(s) ≥ CD sm for all s > 0 with CD > 0, then we can estimate Dε (s) = D(s + ε) ≥ CD (s + ε)m ≥ CD sm

(2.26)

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557

holds with the same values of CD and m, which, notably, are independent of ε. Furthermore, Dε (0) = D(ε) ≥ CD εm > 0. The initial data u0ε ∈ C α (Ω ) for some α ∈ (0, 1), v0ε ∈ C 1 (Ω ), and (u0ε , v0ε ) satisfy u0ε ≥ 0, v0ε ≥ 0, ∥u0ε ∥L∞ (Ω) ≤ ∥u0 ∥L∞ (Ω) + 1, ∥v0ε ∥W 1,∞ (Ω) ≤ ∥v0 ∥W 1,∞ (Ω) + 1, (2.27)

Lemma 2.3. Let f2 ≡ 0, w ≡ 0, τ1 = 1, Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary. Assume that Φε (uε ), φ(vε ) ≡ vε , g(uε ) and f (uε ) satisfy (1.6), (1.8) and (1.9). Suppose that Dε (uε ) satisfies (1.4), and (1.5). Then if satisfies ϵ ≥ 2 and m ≥ 1 − ϵ, system (2.17) has a unique local-in-time classical solution (uε , vε ) such that (uε , vε ) ∈ C 0 ([0, Tmax ); W 1,l (Ω )) ∩ C 2,1 (Ω × [0, Tmax )), where Tmax denotes the maximal existence time. Moreover, if T < +∞ , then ∥uε (·, t)∥L∞ (Ω) + ∥vε (·, t)∥W 1,∞ (Ω) → ∞ Proof . Let U = (uTε vεT )T ∈ R2 . Then, the initial–boundary ⎧ U = ∇ · (F (U )∇U ) + G(U ), ⎪ ⎨ t ∂U = 0, ⎪ ⎩ ∂ν T T U (·, 0) = (uT0ε v0ε ) , where

( F (U ) =

Dε (uε ) −Φε (uε ) 0 1

as t → Tmax .

value problem (2.17) can be reformulated as x ∈ Ω,

t > 0,

x ∈ ∂Ω , x ∈ Ω,

)

( and

t > 0,

G(U ) =

0 −uε g(vε )

) .

Then, applying Theorem 14.4, 14.6 and 15.5 of [40], (2.5) and (2.6) can be proved. Since u0ε ≥ 0 and v0ε ≥ 0, the maximum principle ensures that both uε and vε are nonnegative. 3. Proof of Theorems 2.1 and 2.2 3.1. A priori estimation with τ1 = τ2 = 0 In this subsection, we establish some a priori estimates for solutions to the approximated system (2.1) with non-degenerate diffusion, it is crucial ingredient for the proof of our main results. As a first step towards this, we have the following simple estimate. Lemma 3.1. Let Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary, assume that (uε , vε , wε ) is a classical solution to the system (2.1) on [0, Tmax ). Suppose Φ(uε ), Ψ (uε ), φ(vε ), ψ(wε ) and f (uε ) satisfy (1.6)–(1.8)and D(uε ) satisfies (1.4), and (1.5). Then for all ε ∈ (0, 1), we have ∫ ∫ ∫ d CD (p − m)(p − m − 1) 2 p−2 up−m ≤ − u |∇u | + ϱ(p − m) up−m−1 ε ε ε dt Ω ε 2 Ω Ω ∫ 2 2 2 a χ (p − m)(p − m − 1) |∇vε | p−2+2(α−m) + uε (3.1) 2k1 CD Ω vε ∫ 2 b2 ξ 2 (p − m)(p − m − 1) |∇wε | p−2+2(β−m) + uε 2k2 CD Ω wε with p > 1 + m.

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

558

Proof . Multiplying both sides of the first equation in (2.1) by (p − m)up−m−1 , integrating over Ω , we ε obtain ∫ ∫ d p−m u = (p − m) uεp−m−1 uεt dt Ω ε Ω ∫ = (p − m) uεp−m−1 ∇ · (Dε (uε )∇uε − Φε (uε )∇φ(vε )) Ω ∫ + Ψε (uε )∇ψ(wε ) + (p − m) uεp−m−1 f (uε ) Ω ∫ 2 = −(p − m)(p − m − 1) uεp−m−2 Dε (uε )|∇uε | (3.2) Ω ∫ + a(p − m)(p − m − 1) uεp−m−2 Φε (uε )φ′ (vε )∇uε · ∇vε ∫Ω − b(p − m)(p − m − 1) up−m−2 Ψε (uε )ψ ′ (wε )∇uε · ∇wε ε Ω ∫ + (p − m) uεp−m−1 f (uε ) Ω

By (1.5), we have ∫ − (p − m)(p − m − 1)

2 up−m−2 Dε (uε )|∇uε | ε

∫ ≤ −CD (p − m)(p − m − 1)

Ω

2

up−2 |∇uε | . ε

(3.3)

Ω

And using (1.6), (1.7) and Young’s inequality yield ∫ (p − m)(p − m − 1) up−m−2 Φε (uε )φ′ (vε )∇uε · ∇vε ε Ω∫ ≤ aχ(p − m)(p − m − 1) uεp+α−m−2 φ′ (vε )∇uε · ∇vε Ω ∫ 2 |∇vε | p−2+2(α−m) a2 χ2 (p − m)(p − m − 1) uε ≤ 2k1 CD ∫Ω vε CD (p − m)(p − m − 1) 2 + up−2 |∇uε | . ε 4 Ω

(3.4)

Similarly, we obtain ∫ (p − m)(p − m − 1)

up−m−2 Ψε (uε )ψ ′ (wε )∇uε · ∇wε ε

Ω∫

uεp+β−m−2 ψ ′ (wε )∇uε · ∇wε ∫ 2 |∇wε | p−2+2(β−m) b2 ξ 2 (p − m)(p − m − 1) uε ≤ 2k2 CD Ω wε ∫ CD (p − m)(p − m − 1) 2 + uεp−2 |∇uε | . 4 Ω ≤ bξ(p − m)(p − m − 1)

(3.5)

Ω

Since f satisfies (1.8), we have ∫ (p − m)

up−m−1 f (uε ) ≤ ϱ(p − m) ε

Ω

∫

uεp−m−1

(3.6)

Ω

Combining the above inequalities, the desired result follows. Then we give an estimate on the positivity of the solution components vε and wε , this is the only time in this paper that we require the domain Ω to be convex. The following lemma provides a quantitative estimate on positivity of solutions to the Neumann problem for the Helmholtz equation with nonnegative inhomogeneity having given norm in L1 (Ω ).

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

Lemma 3.2 ([41]). Let ϑ ∈ C 0 (Ω ) be a nonnegative function such that to { −△z + z = ϑ, x ∈ Ω , ∂z x ∈ ∂Ω , ∂ν = 0, then z ≥ ρ0 :=

(∫

∞

1 N

0

e−(t+

(diamΩ)2 ) 4t

∫ Ω

559

ϑ > 0. If z is a classical solution

) ∫ dt · ϑ > 0 in Ω .

(4πt) 2

Ω

Lemma 3.3. Let the assumptions in Lemma 2.1 hold, (uε (·, t), vε (·, t), wε (·, t)) be a solution to (2.1). Then there exists a constant C > 0, for any ε > 0, we have ∥uε (·, t)∥L1 (Ω) ≤ C

f or all t ∈ [0, Tmax ),

∥vε (·, t)∥L1 (Ω) ≤ C

f or all t ∈ [0, Tmax ),

∥wε (·, t)∥L1 (Ω) ≤ C

f or all t ∈ [0, Tmax ).

Proof . Then proof is similar to [27], to avoid repetition, so we omitted it. In addition, on the right of (3.1), the third and the fourth term can be controlled by the first dissipative one under proper conditions. This is showed in the following lemma. Lemma 3.4. Let the assumptions in Lemma 3.1 hold. (1) Assume that k1 < 1, k2 < 1, α − m < 1 − (1 − N2 )(1 − k1 ), β − m < 1 − (1 − Let p > 0 satisfy

2 N )(1

− k2 ) and N ≥ 3.

{ (1 − 2 )(1 − 2k ) 2 1 N p > max , 1 − , 4 − 4(α − m), 5 − 4(α − m), 4 − 4(β − m), 5 − 4(β − m), N 1 + N2 (1 − 2k1 ) (1 − N2 )(1 − 2k2 ) 2 1 1 2 , (1 − )θ, (1 − )υ, (N − 2)(α − m − ), (N − 2)(β − m − ), 2 N N 2 2 1 + N (1 − 2k2 ) (N − 2)(α − m − k1 ) 1 − (α − m) (N − 2)(β − m − k2 ) 1 − (β − m) } , , , , 2(1 − k1 ) 1 − k1 2(1 − k2 ) 1 − k2 and any ε > 0 ∫ Ω

2

|∇vε | vε2k1

Ω

∫

2

∀t ∈ (0, Tmax )

2

∀t ∈ (0, Tmax )

uεp−2 |∇uε | + C,

Ω

|∇wε |

∫

up−2+2(α−m) ≤ε ε 2

up−2+2(β−m) ε 2k2 wε

∫ ≤ε

up−2 |∇uε | + C, ε

Ω

∫ (1−k1 )p (1−k2 )p where θ = 1−(α−m) , υ = 1−(β−m) , C > 0 determined by p, α − m, β − m, k1 , k2 , ρ0 , Ω , ε on Ω u0 . (2) Assume that k1 ≥ 1, k2 ≥ 1, if α − m < 1, β − m < 1, then for any p > p3 := {4 − 4(α − m), 4 − 4(β − m), 1}, and any ε > 0, ∫ Ω

∫ Ω

2

|∇vε |

up−2+2(α−m) ≤ε ε

vε2k1 |∇wε | wε2k2

∫

2

∀t ∈ (0, Tmax )

2

∀t ∈ (0, Tmax )

uεp−2 |∇uε | + C,

Ω 2

up−2+2(β−m) ≤ε ε

∫

up−2 |∇uε | + C, ε

Ω

where C > 0 determined by p, α − m, β − m, k1 , k2 , ρ0 , Ω , ε on

∫ Ω

u0 .

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

560

p−2+2(α−m)

uε

Proof . (1) Multiply the second equation of (2.1) by obtain |∇vε |

∫ (2k1 − 1)

with p > 0 and integrate by parts to

2k −1 vε 1

2

up−2+2(α−m) − (p − 2 + 2(α − m)) ε

2k1 Ω vε ∫ p−2+2(α−m) uε −α1 vε2k1 −2 Ω

∫

p−3+2(α−m)

uε

∇uε · ∇vε

vε2k1 −1

Ω p−2+2(α−m)

∫

uε uε

+ β1

= 0.

vε2k1 −1

Ω

Now, we divided the proof into three cases. (i) k1 < 12 Due to the non-negativity of uε and vε , and 2k1 − 1 < 0, we have ∫ Ω

∫ p−3+2(α−m) p − 2 + 2(α − m) uε |∇uε | · |∇vε | 1 − 2k1 vε2k1 −1 Ω ∫ p−1+2(α−m) uε β1 + 1 − 2k1 Ω vε2k1 −1 =: H1 + H2 .

2

|∇vε | vε2k1

up−2+2(α−m) ≤ − ε

(3.7)

Let p > p1 := 4 − 4(α − m) and ε > 0. As regards H1 , by the Cauchy inequality, we have H1 ≤

ε 8

∫

2

up−2 |∇uε | + c1 ε

∫

Ω

Ω

2

|∇vε |

vε4k1 −2

up−4+4(α−m) ε

(3.8)

with some positive constant c1 determined by p, α − m, k1 and ε. Since α − m < 1 − (1 − we make use of the Young’s inequality to obtain ∫ c1

|∇vε |

4k1 Ω vε

2

up−4+4(α−m) ≤ −2 ε

1 2

2

|∇vε |

∫ Ω

vε2k1

uεp−2+2(α−m) + c2

∫

2 N )(1

(1−k1 )p

2

|∇vε | · vε1−(α−m)

− k1 ) < 1,

−2

(3.9)

Ω

with some positive constant c2 relying on p, α − m and c1 . Firstly, we claim that there exists constant c3 > 0 such that ∫ ∫ (1−k1 )p −2 ε 2 2 1−(α−m) ≤ c2 |∇vε | · vε up−2 |∇uε | + c3 (3.10) ε 8 Ω Ω ∫ (1−k1 )p where c3 depending on p, α−m, k1 , Ω , ε on Ω u0 . Denoted θ = 1−(α−m) , due to α−m < 1−(1− N2 )(1−k1 ) < 1 and p > (1 − N2 )θ, it is readily to know that θ > 1. Testing the second equation of (2.1) by vεθ−1 and using the H¨ older inequality, Sobolev embedding theorem, elliptic regular arguments yield that ∫ ∫ ∫ 1 1 2 |∇vε | vεθ−2 = uε vεθ−1 − vεθ θ − 1 θ − 1 Ω Ω Ω 1 θ−1 ≤ ∥vε ∥L(θ−1)ι (Ω) ∥uε ∥Lι′ (Ω) θ−1 c4 ≤ ∥vε ∥θ−1 N (θ−1)ι ∥uε ∥Lι′ (Ω) (3.11) 2, θ−1 W N +2(θ−1) (Ω) c5 ≤ ∥uε ∥θ−1N (θ−1)ι ∥uε ∥Lι′ (Ω) θ−1 N +2(θ−1) L

(Ω)

N , 1} < ι < ∞, 1ι + ι1′ = 1, c4 , c5 > 0 are constants determined by Ω and (θ − 1)ι. By where max{ (N −2)(θ−1) the Gagliardo–Nirenberg inequality, there exist c6 , c7 > 0 such that

∥uε ∥θ−1N (θ−1)ι L N +2(θ−1) (Ω)

p

2(θ−1) p 2N (θ−1)ι L p(N +2(θ−1)) (Ω)

= ∥uε2 ∥

( p p ≤ c6 ∥∇uε2 ∥cL2 (Ω) ∥uε2 ∥1−c p

L 2 (Ω)

p

+ c6 ∥uε2 ∥

p L 2 (Ω)

) 2(θ−1) p

(3.12)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

and

2

p

( p p ≤ c7 ∥∇uε2 ∥dL2 (Ω) ∥uε2 ∥1−d 2

∥uε ∥Lι′ (Ω) = ∥uε2 ∥ p 2ι′

L p (Ω)

L p (Ω)

with c=

) p2

p

+ c7 ∥uε2 ∥

561

(3.13)

2 L p (Ω)

N p(θ − 1)ι − p(N + 2(θ − 1)ι) ∈ (0, 1), (N p + 2 − N )(θ − 1)ι d=

N p(ι′ − 1) ∈ (0, 1) + 2 − N)

ι′ (N p

+ d × p2 < 2 due to p > (1 − N2 )θ. Therefore, together with (3.11)–(3.13), Lemma 3.3 satisfying c × 2(θ−1) p and Young’s inequality, (3.10) is obtained. According to (3.8)–(3.10), we have ∫ ∫ 2 ε 1 |∇vε | p−2+2(α−m) 2 H1 ≤ up−2 |∇u | + u + c3 . (3.14) ε 4 Ω ε 2 Ω vε2k1 ε Secondly, we claim that there exist c8 > 0 such that β1 H2 = 1 − 2k1

p−1+2(α−m)

∫

uε

ε ≤ 4

vε2k1 −1

Ω

∫

2

up−2 |∇uε | + c8 ε

(3.15)

Ω

∫ with c8 relying on p, α − m, k1 , Ω , ε on Ω u0 . In fact, since p > p1 , fixed p > 2 − 2(α − m), and ε > 0, due to α − m < 1 − (1 − N2 )(1 − k1 ), we have 2 − 2(α − m) ≥ (N − 2)(α − m − k1 ) and hence { (N − 2)(α − m − k ) 1 (1 − 1 p > max , (N − 2)(α − m − ), 2(1 − k1 ) 2 1+

2 N )(1 − 2k1 ) , 2 N (1 − 2k1 )

1−

2} . N

By the H¨ older inequality, Sobolev embedding theorem, elliptic regularity arguments, we have ∫

p−1+2(α−m)

uε

Ω

1 ≤ ∥vε ∥1−2k (1−2k1 )σ

vε2k1 −1

L

p−1+2(α−m)

(Ω)

∥uε ∥

′

L(p−1+2(α−m))σ (Ω)

≤ c9 ∥vε ∥1−2k1(1−2k1 )N σ

∥uε ∥

2,

W N +2(1−2k1 )σ (Ω) 1−2k1

≤ c10 ∥uε ∥

(1−2k1 )N σ L N +2(1−2k1 )σ (Ω)

p−1+2(α−m)

p−1+2(α−m)

∥uε ∥

′

L(p−1+2(α−m))σ (Ω)

with some positive constants c9 , c10 depending on Ω , (1 − 2k1 )σ, σ1 + σ1′ = 1. Denoted n := the Gagliardo–Nirenberg inequality, there exist c11 , c12 > 0 such that p

1 2 ∥uε ∥1−2k Ln (Ω) = ∥uε ∥

2(1−2k1 ) p 2n L p (Ω)

(3.16)

′

L(p−1+2(α−m))σ (Ω)

( p p ≤ c11 ∥∇uε2 ∥lL2 (Ω) ∥uε2 ∥1−l 2

L p (Ω)

1) ) 2(1−2k p

p

+ c11 ∥uε2 ∥

(1−2k1 )N σ N +2(1−2k1 )σ ,

2 L p (Ω)

by

(3.17)

and p

p−1+2(α−m)

∥uε ∥

′ L(p−1+2(α−m))σ (Ω)

2(p−1+2(α−m)) p 2(p−1+2(α−m))σ ′ p L (Ω)

= ∥uε2 ∥ ≤

(

p 2

l= r=

p

c12 ∥∇uε ∥rL2 (Ω) ∥uε2 ∥1−r 2 L p (Ω)

with

(3.18) p 2

+ c12 ∥uε ∥

) 2(p−1+2(α−m)) p 2

L p (Ω)

N p(n − 1) ∈ (0, 1) n(N p + 2 − N )

N p(p − 2 + 2(α − m)) ∈ (0, 1) (N p + 2 − N )(p − 1 + 2(α − m))

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

562

1) + r × 2(p−1+2(α−m)) < 2 due to α − m < 1 − (1 − N2 )(1 − k1 ). By means of (3.16)–(3.18), satisfying l × 2(1−2k p p Lemma 3.3 and Young’s inequality, (3.14) is obtained. According to (3.7), (3.14) and (3.15), the first desired results are proved. Similarly, we can conclude that

|∇wε |

∫

2

up−2+2(β−m) ≤ε ε

wε2k2

Ω

∫

2

uεp−2 |∇uε | + C,

∀t ∈ (0, Tmax ).

Ω

(ii) k1 = 21 Since α − m < 1 − (1 − N2 )(1 − k1 ), one can choose some k˜1 < 12 such that α − m < 1 − (1 − N2 )(1 − k˜1 ). Then it follows from Lemma 3.2 and case (i) that for any p > p1 and any ε > 0 2

∫ Ω

|∇vε | p−2+2(α−m) uε = vε

|∇vε |

∫

2

Ω

˜

vε2k1 −1 up−2+2(α−m) ε

˜ vε2k1

∫ 2 |∇vε | p−2+2(α−m) ˜ ≤ ρ20k1 −1 uε ˜ 2k Ω vε 1 ∫ 2 ≤ε uεp−2 |∇uε | + c13 Ω

with some positive constant c13 determined by p, α − m, k˜1 , Ω , ε on proved. Taking a similar procedure, we can conclude that 2

∫ Ω

(iii)

1 2

|∇wε | p−2+2(β−m) uε ≤ε wε

∫

2

up−2 |∇uε | + C, ε

∫ Ω

u0 , the first desired results are

∀t ∈ (0, Tmax ).

Ω

< k1 < 1 Due to the non-negativity of uε and vε , and 2k1 − 1 > 0, we have

∫ p−3+2(α−m) p − 2 + 2(α − m) uε |∇uε | · |∇vε | 2k1 2k1 − 1 vε2k1 −1 Ω vε Ω ∫ p−2+2(α−m) α1 uε + 2k1 − 1 Ω vε2k1 −2 =: K1 + K2 . } { and ε > 0. By the Cauchy inequality, we can get Let p > p2 := max 5 − 4(α − m), 1−(α−m) 1−k1 ∫

2

|∇vε |

up−2+2(α−m) ≤ ε

ε K1 ≤ 8

∫

2 up−2 |∇uε | ε

∫ + c14

Ω

Ω

(3.19)

2

|∇vε |

vε4k1 −2

up−4+4(α−m) ε

with some positive constant relying on p, α − m, k˜1 and ε. Owing to α − m < 1 − (1 − N2 )(1 − k1 ), we utilize the Young’s inequality to obtain ∫ c14 Ω

|∇vε | vε4k1

2

up−4+4(α−m) ≤ −2 ε

1 2

∫

2

|∇vε |

uεp−2+2(α−m) + c15

vε2k1

Ω

∫

2

(1−k1 )p −2

|∇vε | · vε1−(α−m)

Ω

with some positive constant c15 relying on p, α − m and c14 . Similar to (3.10), we have ∫

2

(1−k1 )p

|∇vε | · vε1−(α−m)

c15

−2

≤

Ω

where c16 > 0 depending on p, α − m, k1 , Ω , ε on K1 ≤

ε 4

∫ Ω

2

up−2 |∇uε | + ε

∫ Ω

1 2

ε 8

∫

2

up−2 |∇uε | + c16 ε

Ω

u0 , for α − m < 1 − (1 −

∫ Ω

2 N )(1

− k1 ). Therefore

2

|∇vε | vε2k1

uεp−2+2(α−m) + c16 .

(3.20)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

563

Moreover, similar to (3.15), we have K2 =

β1 1 − 2k1

∫

p−1+2(α−m)

uε

Ω

vε2k1 −1

≤

ε 4

∫

2

up−2 |∇uε | + c17 ε

(3.21)

Ω

∫ with c17 > 0 relying on p, α−m, k1 , Ω , ε on Ω u0 , since α−m < 1−(1− N2 )(1−k1 ) < 1−(1− N2 )(1−k1 )+ N1 . Combining with (3.19)–(3.21), the first desired conclusion is obtained. Similarly, we can get that ∫ ∫ 2 |∇wε | p−2+2(β−m) 2 uε ≤ε uεp−2 |∇uε | + C, ∀t ∈ (0, Tmax ). 2k2 Ω wε Ω p−1+2(α−m)

and (2) Let p > {4 − 4(α − m), 1} and ε > 0. Multiplying the second equation of (2.1) by uε vε integrating by parts give rise to ∫ ∫ ∫ p−3+2(α−m) 2 |∇vε | p−2+2(α−m) uε uε ≤ (p − 2 + 2(α − m)) |∇uε | · |∇vε | + α1 uεp−2+2(α−m) vε2 vε Ω Ω Ω ∫ ∫ p−3+2(α−m) uε |∇uε | · |∇vε | + α1 upε (3.22) ≤ (p − 2 + 2(α − m)) vε Ω Ω =: R1 + R2 . As regards R1 , by the Young’s inequality, we have ∫ ∫ 2 ε |∇vε | p−4+4(α−m) 2 p−2 R1 ≤ 2−2k uε |∇uε | + c18 uε vε2 4ρ0 1 Ω Ω ∫ ∫ ∫ 2 2 1 ε |∇vε | p−2+2(α−m) |∇vε | 2 up−2 |∇u | + ≤ 2−2k u + c ε 19 ε ε 2 Ω vε2 vε2 4ρ0 1 Ω Ω

(3.23)

where c18 , c19 > 0 depending on p, α − m, k1 , ρ0 and ε. Testing the second equation of (2.1) by vε−1 and integrating by parts, we get ∫ ∫ 2 uε |∇vε | = |Ω | − ≤ |Ω |. (3.24) 2 v Ω vε Ω ε By the Gagliardo–Nirenberg inequality, there exist c20 > 0 such that ∫ ( )2 p p p p 1 2 R2 = α1 + c ∥u ∥ upε = α1 ∥(uε + ε) 2 ∥2L2 (Ω) ≤ c20 ∥∇uε2 ∥rL12 (Ω) ∥uε2 ∥1−r 2 ε 20 2 L p (Ω)

Ω

L p (Ω)

with

N (p − 1) ∈ (0, 1). Np + 2 − N Consequently, by the Young’s inequality, we have ∫ ε 2 R2 ≤ 2−2k uεp−2 |∇uε | + c21 4ρ0 1 Ω r1 =

(3.25)

with some positive constant c21 determined by p, α−m, k1 , ρ0 and ε. According to (3.22)–(3.25), we conclude ∫ ∫ 2 |∇vε | p−2+2(α−m) ε 2 uεp−2 |∇uε | + c22 uε ≤ 2−2k 2 1 v ρ0 Ω Ω ε with c22 := 2(c19 |Ω | + c21 ), combining with Lemma 3.2 results in ∫ ∫ ∫ 2 2 |∇vε | p−2+2(α−m) |∇vε | p−2+2(α−m) 2 2−2k1 1 u ≤ ρ u ≤ ε uεp−2 |∇uε | + c22 ρ2−2k , ε ε 0 0 2k1 vε2 Ω vε Ω Ω 1 taking C = c22 ρ2−2k , the first desired results are obtained. Similarly, we can get that 0 ∫ ∫ 2 |∇wε | p−2+2(β−m) 2 u ≤ ε uεp−2 |∇uε | + C, ∀t ∈ (0, Tmax ). ε 2k2 Ω wε Ω

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Lemma 3.5. Let the assumptions in Lemmas 3.1 and 3.4 hold. We claim that for any p > max{1 + m, p1 , ( N2 − 1)β, 1 − N2 }, there exists C > 0 such that ∥uε (·, t)∥Lp−m (Ω) ≤ C

(3.26)

for all t ∈ (0, Tmax ), where C is determined by p. Proof . By Lemmas 3.1 and 3.4, we have ∫ ∫ ∫ d CD (p − m)(p − m − 1) 2 p−2 up−m ≤ − u |∇u | + ϱ(p − m) up−m−1 ε ε ε dt Ω ε 2 Ω Ω ∫ ( ) 2 2 2 a χ (p − m)(p − m − 1) CD 2 up−2 |∇uε | + c1 + CD 8a2 χ2 Ω ε ) 2 ∫ b2 ξ 2 (p − m)(p − m − 1) ( CD 2 p−2 u |∇u | + c + ε 2 ε CD 8b2 ξ 2 Ω with some positive constants c1 , c2 depending on p, k1 , k2 , α − m, β − m, Ω , d dt

∫

up−m ε

Ω

2 2

with c3 = c1 a

CD (p − m)(p − m − 1) ≤− 4

χ (p−m)(p−m−1) CD

2 2

+ c2 b

∫

ξ (p−m)(p−m−1) . CD

∫ ϱ(p − m) Ω

2 up−2 |∇uε | ε

1 , 1 8a2 χ2 8b2 ξ 2

∫ + ϱ(p − m)

Ω

and

∫ Ω

uεp−m−1 + c3

u0 , that is (3.27)

Ω

By the Young’s inequality, we have

up−m−1 ≤ ϱ(p − m − 1) ε

∫ Ω

p−m

up−m + ε

|Ω | . p−m

(3.28)

Taking (3.28) into (3.27), we have ∫ ∫ ∫ d CD (p − m)(p − m − 1) 2 p−2 up−m ≤ − u |∇u | + ϱ(p − m − 1) uεp−m + c4 ε ε dt Ω ε 4 Ω Ω p−m

with c4 = c3 + |Ω| p−m . According to a straightforward ordinary differential equation comparison argument, we obtain the desired results. Next, we shall prove the global boundedness of solutions for (2.1). To the best of our knowledge, there are two well-established methods to establish a uniform boundedness on uε in the literatures. One method to derive such L∞ -boundedness is based on the Moser–Alikakos iterative technique in Lemma A.1 of [29], and the another way is dependent upon a series of standard semigroup arguments. In this paper, we apply the former one in the proof of the following lemma, we can extend the local-in-time solution to the global-in-time solution by virtue of Lemmas 2.1 and 3.5, which allows for an extension of the outcome in Lemma 3.5 from [0, Tmax ) to [0, ∞). Lemma 3.6. Let the assumptions in Lemmas 3.1, 3.4 and 3.5 { } hold. If there exists some p − m > p˘ := N (1+m)+2(1+2m) (N −2)m , such that uε ∈ L∞ (0, Tmax ; Lp (Ω )), then N − m, (N + 2)α+ − m, γ0 − m, 1 − 2 2 ∥uε (·, t)∥L∞ (Ω) ≤ C for all t > 0, with some positive constant C, where α+ = max{α, 0}, γ0 := inf {γ ≥ 0, h(p) > 0 f or all p ∈ (γ, ∞)}, h(p) = p2 + (N + mN + m − m(N + 2)α+ )p − m(N + 2)α+ .

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Proof . Since p > N , by the Sobolev embedding theorem and elliptic regularity arguments, we have ∥vε (·, t)∥W 1,∞ (Ω) ≤ c1 ∥vε (·, t)∥W 2,p (Ω) ≤ c2 ∥uε (·, t)∥Lp (Ω) ≤ c2 ∥uε ∥L∞ (0,Tmax ;Lp (Ω)) < ∞, similar to vε , we have ∥wε (·, t)∥W 1,∞ (Ω) < ∞ for all t ∈ (0, Tmax ) with some positive constants c1 , c2 determined by Ω and p, which together with (1.6) and Lemma 2.1 guarantee that φ′ (vε )∇vε , ψ ′ (wε ) ∇wε ∈ L∞ (∞ × (0, Tmax )). On account of Φε (uε ), Ψε (uε ) satisfies (1.5) and uε ∈ L∞ (0, Tmax ; Lp (Ω )), it is easy to check that g(x, t) := −Φε (uε )φ′ (vε )∇vε + Ψε (uε )ψ ′ (wε )∇wε ∈ L∞ (0, Tmax ; Lq (Ω )), with q = αp+ . Moreover, because of the nonnegativity of uε , f (uε ) ≤ ϱ. All necessary conditions for p in Lemma A.1 of [29] are also satisfied, we chose p sufficiently large such that p − m > p˘, and hence supt∈[0,Tmax ) ∥uε (·, t)∥L∞ (Ω) < ∞, the desired results are proved. Thus, we obtain that (uε , vε , wε ) is the global bounded classical solution to the approximated system (2.1). Finally, we prove the main theorem. Proof of Theorem 2.1. Firstly, with the help of Lemma 3.6 we derive that for each ε ∈ (0, 1), system (2.1) admits a classical solution (uε , vε , wε ) which is defined for all t > 0. Let ϕ ∈ W0k,2 (Ω ) with 2k > N , it is known by the embedding theorem, we see that W0k,2 (Ω ) ↪→ L∞ (Ω ). Thus, ϕ ∈ L∞ (Ω ) and ∥ϕ∥L∞ (Ω) ≤ c1 ∥ϕ∥W k,2 (Ω) with some positive constant c1 . Multiplying both sides of the first equation in 0

ϕ and integrating by parts on Ω , we obtain (2.1) by (p − m)up−m−1 ε ∫ ( ∫ ∂ p−m ) uε ϕ = (p − m) up−m−1 uεt ϕ (3.29) ε Ω ∂t ∫ Ω ∫ = (p − m) up−m−1 ∇ · (Dε (uε )∇uε − Φε (uε )∇φ(vε ) + Ψε (uε )∇ψ(wε ))ϕ + (p − m) f (uε )ϕ ε Ω

Ω

Using H¨ older’s and Young’s inequalities, the steps are similar to the proof of the Lemmata 3.1, 3.4 and 3.5, we obtain ∫ ( ( b2 ξ 2 (p − m)(p − m − 1) ∫ |∇w |2 ∂ p−m ) ε uε ϕ ≤ c1 uεp−2+2(β−m) 2k2 CD Ω wε Ω ∂t ∫ 2 |∇vε | p−2+2(α−m) a2 χ2 (p − m)(p − m − 1) uε + 2k1 CD Ω vε ∫ CD (p − m)(p − m − 1) 2 +− uεp−2 |∇uε | 2 Ω ∫ ) + ϱ(p − m) up−m−1 ∥ϕ∥|W k,2 (Ω) . ε 0

Ω

Using Young’s inequality, we have ∫ ( ( b2 ξ 2 (p − m)(p − m − 1) ∫ |∇w |2 ∂ p−m ) ε uε ϕ ≤ c1 up−2+2(β−m) ε 2k2 ∂t C D Ω Ω wε ∫ 2 a2 χ2 (p − m)(p − m − 1) |∇vε | p−2+2(α−m) + uε 2k1 CD ∫ Ω vε CD (p − m)(p − m − 1) 2 − up−2 |∇uε | ε 2 Ω ∫ ) p−m + ϱ(p − m − 1) uε + c2 ∥ϕ∥|W k,2 (Ω) . Ω

0

(3.30)

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G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

This yields that ( b2 ξ 2 (p − m)(p − m − 1) ∫ |∇w |2 ∂ p−m ε ∥ uε ∥(W k,2 (Ω))∗ ≤ c1 uεp−2+2(β−m) 2k2 ∂t CD 0 Ω wε ∫ 2 a2 χ2 (p − m)(p − m − 1) |∇vε | p−2+2(α−m) + uε 2k1 CD ∫ Ω vε CD (p − m)(p − m − 1) 2 − up−2 |∇uε | ε 2 Ω ∫ ) p−m + ϱ(p − m − 1) uε + c2 Ω

Integrating above inequality with respect to t, we obtain ∂ p−m u ∥L1 (0,T ;(W k,2 (Ω))∗ ) ∂t ε 0 ( b2 ξ 2 (p − m)(p − m − 1) ∫ T ∫ |∇w |2 ε ≤ c1 uεp−2+2(β−m) 2k2 CD 0 Ω wε ∫ ∫ 2 a2 χ2 (p − m)(p − m − 1) T |∇vε | p−2+2(α−m) uε + 2k1 CD 0 Ω vε ∫ ∫ CD (p − m)(p − m − 1) T 2 up−2 |∇uε | − ε 2 0 Ω ∫ T∫ ) + ϱ(p − m − 1) up−m + c T 2 ε ∥

0

(3.31)

Ω

Thus, there exists some positive constant c3 such that ∥

∂ p u ∥ ≤ c3 f or all t ∈ (0, T ). k,2 ∂t ε L1 (0,T ;(W0 (Ω))∗ )

(3.32)

Let ε ∈ (0, 1), T ∈ (0, ∞), and p > p˘. Then from Lemma 3.6, we have ∥uε (·, t)∥L∞ ((0,∞);L∞ (Ω)) ≤ C holds as well as ∥vε (·, t)∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ C, and ∥wε (·, t)∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ C, where C > 0 is independent of T and ε. Then there exist three subsequences {uεi }i∈N , {vεi }i∈N and {wεi }i∈N and three functions u ∈ L∞ ((0, ∞); L∞ (Ω )), v ∈ L∞ ((0, ∞); W 1,∞ (Ω )) and w ∈ L∞ ((0, ∞); W 1,∞ (Ω )) such that ⎧ ∗ ∞ ∞ ⎪ ⎨uεi ⇀ u weakly in L ((0, ∞); L (Ω )), (3.33) vεi ⇀ v weakly ∗ in L∞ ((0, ∞); W 1,∞ (Ω )), ⎪ ⎩ wεi ⇀ w weakly ∗ in L∞ ((0, ∞); W 1,∞ (Ω )), ∫s where εi → 0 as i → ∞. Define G(s) := 0 Dε (ρ)dρ and ∥uε (·, t)∥L∞ ((0,∞);L∞ (Ω)) ≤ C, there exists c5 > 0 such that |∇G(uε )| ≤ c5 |∇um+1 |, it is easy to know ∇um+1 is bounded in L2 ((0, T ); L2 (Ω )). Hence, ∇G(uε ) ε ε 2 2 2 is bounded in L ((0, T ); L (Ω )), and then there exist ϑ ∈ L ((0, ∞); L2 (Ω )) and a subsequence (still denoted by {uεi }i∈N ) such that ∇G(uε ) ⇀ ϑ weakly in

L2loc ((0, ∞); L2 (Ω )) as i → ∞.

(3.34)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

567

Similarly ∇vεi ⇀ ∇v weakly in

L2 ((0, ∞); L2 (Ω )) as i → ∞

(3.35)

∇wεi ⇀ ∇v weakly in

L2 ((0, ∞); L2 (Ω )) as i → ∞.

(3.36)

and ∂ p−m uε is bounded in We know from the proof above that up−m is bounded in L2 ((0, T ); W 1,2 (Ω )) and ∂t ε k,2 1 ∗ L (0, T ; (W0 (Ω )) ) for any T ∈ (0, ∞), respectively. It is known by the embedding theorem and Aubin– Lions theorem that there exists a subsequence (still denoted by {uεi }i∈N ) such that

up−m → up−m εi

L2 ((0, T ); L2 (Ω )) as i → ∞.

strongly in

From the Riesz lemma and diagonal argument, the L2 convergence implies that there exists a subsequence (still denoted by {uεi }i∈N ) such that up−m → up−m εi

a.e. on

Ω × (0, ∞) as i → ∞.

Due to p > m + 1, we obtain uεi → u

Ω × (0, ∞) as i → ∞.

a.e. on

(3.37)

It is known by the embedding theorem and Aubin–Lions theorem that there exists a subsequence (still denoted by {vεi }i∈N ) such that vε i → v

L2 ((0, T ); L2 (Ω )) as i → ∞

strongly in

and then vε i → v

Ω × (0, ∞) as i → ∞.

a.e. on

(3.38)

Similarly, we have wεi → w

L2 ((0, T ); L2 (Ω )) as i → ∞

strongly in

and then w εi → w

a.e. on

Ω × (0, ∞) as i → ∞.

(3.39)

Using (3.37)–(3.39), we obtain Φεi (uεi )∇φ(vεi ) → Φ(u)∇φ(v) a.e. on

Ω × (0, ∞) as i → ∞.

(3.40)

and Ψεi (uεi )∇ψ(wεi ) → Ψ (u)∇ψ(w) a.e. on

Ω × (0, ∞) as i → ∞.

(3.41)

due to the boundedness of uε , vε and wε . Thus, by dominated convergence theorem and diagonal, we can obtain that there exists a subsequence (still denoted by {εi }i∈N ) such that Φεi (uεi )∇φ(vεi ) → Φ(u)∇φ(v) strongly in Ψεi (uεi )∇ψ(wεi ) → Ψ (u)∇ψ(w) strongly in

L2 ((0, T ); L2 (Ω )) as i → ∞. L2 ((0, T ); L2 (Ω )) as i → ∞.

(3.42) (3.43)

For any given T ∈ (0, ∞), we take ϕ ∈ C0∞ (Ω × [0, T )). Then multiplying system (2.1) by ϕ and integrating on Ω × (0, T ), we can see that ∫ T∫ ∫ ∫ T∫ ∫ T∫ uεi ϕt − u0εi ϕ(·, 0) = − Dε (uεi )∇uεi · ∇ϕ + Φ(uεi )∇φ(vεi ) · ∇ϕ 0 Ω Ω 0 ∫ Ω∫ ∫0 T ∫Ω T + Ψ (uεi )∇ψ(wεi ) · ∇ϕ + f (uεi )ϕ (3.44) 0

Ω

0

Ω

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

568

holds as well as

T

∫

∫

T

∫

∫

and

T

∫

Ω

0

∫

T

∫

∫

uεi ϕ

(3.45)

uεi ϕ.

(3.46)

Ω

0

∫

∇wεi ∇ϕ + α2

T

∫

wεi ϕ = β2

Ω

0

∫

vε i ϕ = β1

Ω

0

T

∫

∇vεi ∇ϕ + α1

Ω

0

0

Ω

By (3.37), we see that G(uεi )△ϕ → G(u)△ϕ

Ω × (0, ∞) as i → ∞.

a.e. on

Moreover, we have |G(uεi )△ϕ| ≤ (C + 1)∥D∥L∞ (0,C+1) ∥△ϕ∥L∞ (Ω) . Thus, by the dominated convergence theorem, we obtain that ∫ T∫ ∫ T∫ G(uεi )△ϕ → G(u)△ϕ as i → ∞. (3.47) Ω

0

Ω

0

From (3.34), we see that T

∫

∫

T

∫

∫

∇G(uεi ) · ∇ϕ →

ϑ · ∇ϕ as i → ∞.

Ω

0

Then (3.47) and (3.48) imply that ϑ = ∇G(u), and thus, we have ∫ T∫ ∫ T∫ ∇G(uεi ) · ∇ϕ → ∇G(u) · ∇ϕ as i → ∞. Ω

0

Ω

T

∫

Ω

T

T

∫

∫

T

T

∫

∫

Ω

∇v · ∇ϕ as i → ∞.

(3.52)

∇w · ∇ϕ as i → ∞.

(3.53)

uϕt as i → ∞.

(3.54)

∫

∇wεi · ∇ϕ → 0

(3.51)

Ω

0

∫

Ψ (u)∇ψ(v) · ∇ϕ as i → ∞. Ω

∇vεi · ∇ϕ → Ω

0

(3.50)

∫

0

From (3.35) and (3.36), we obtain ∫ T∫

Φ(u)∇φ(v) · ∇ϕ as i → ∞. Ω

0

Similarly, using (3.36) and (3.43) ∫ T∫ ∫ Ψεi (uεi )∇ψ(vεi ) · ∇ϕ → 0

(3.49)

Ω

0

Using (3.35) and (3.42), we have ∫ T∫ ∫ Φεi (uεi )∇φ(vεi ) · ∇ϕ → 0

(3.48)

Ω

0

Ω

0

From (3.33), we have ∫

T

∫

∫

T

∫

uεi ϕt → 0

Ω

0

Ω

Similarly, since f (u) is smooth and ∥uε ∥L∞ (Ω) ≤ C, we obtain that ∥f (uε )ϕ∥L∞ (Ω) ≤ c4 with some positive constant c4 , and thus f (uε )ϕ ∈ L2 ((0, T ); L2 (Ω )). therefore, from the strongly convergence of uε , we find that f (uε )ϕ → f (u)ϕ strongly in L2 ((0, T ); L2 (Ω )) as i → ∞. (3.55) Finally, according to the construction of the initial data, we obtain ∫ ∫ ∫ ∫ u0εi ζ(·, 0) → u0 ζ(·, 0) as i → ∞, v0εi ζ(·, 0) → v0 ζ(·, 0) as i → ∞. Ω

Ω

Ω

Ω

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

∫

569

∫ w0εi ζ(·, 0) →

Ω

w0 ζ(·, 0) as i → ∞

(3.56)

Ω

Thus, by using (3.49)–(3.56), we can pass to the limit in each term of the identities (3.44)–(3.46) and obtain T

∫

∫

∫

0

Ω

T

∫

uϕt −

−

∫

Ω

Ω

0

∫

T

T

Ψ (u)∇ψ(w) · ∇ϕ +

∫

∫

T

0

∫

∇v∇ϕ + α1 Ω

0

and ∫

T

∫

T

uϕ Ω

0

∫

∫

T

∫

wϕ = β2 0

Ω

Ω

∫

vϕ = β1 ∫

Ω

T

∫

Ω

0

∇w∇ϕ + α2 0

Φ(u)∇φ(v) · ∇ϕ ∫ f (u)ϕ

Ω ∫ T

0

Ω

0

∫

∫

∫

+ holds as well as

T

∫ D(u)∇u · ∇ϕ +

u0 ϕ(·, 0) = −

uϕ 0

Ω

Hence, (u, v, w) is a global weak solution to system (1.2). Moreover, we can deduce from (3.33) and Lemma 3.6 that ∥u∥L∞ ((0,T );L∞ (Ω)) ≤ lim inf ∥uε ∥L∞ ((0,T );L∞ (Ω)) ≤ C, i→∞

∥v∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ lim inf ∥vε ∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ C, i→∞

∥w∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ lim inf ∥wε ∥L∞ ((0,∞);W 1,∞ (Ω)) ≤ C, i→∞

which gives the boundedness of (u, v, w). Thus, we complete the proof of Theorem 2.1. 3.2. A priori estimation with τ1 = 1, τ2 = 0 In this subsection, we will establish some priori estimates for solution to the approximated system (2.1) with τ1 = 1, τ2 = 0 for special choice Φ(uε ) = χuε , the system reads as ⎧ uεt = ∇ · (Dε (uε )∇uε − χuε ∇φ(vε ) + Ψε (uε )∇ψ(wε )) + f (uε ), x ∈ Ω , t > 0, ⎪ ⎪ ⎪ ⎪ v = △v − α v + β u , x ∈ Ω , t > 0, εt ε 1 ε 1 ε ⎪ ⎨ 0 = △wε − α2 wε + β2 uε , x ∈ Ω , t > 0, (3.57) ∂uε ∂vε ∂wε ⎪ ⎪ ⎪ = 0, = 0, = 0, x ∈ ∂Ω , t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ ∂ν uε (x, 0) = u0ε (x), vε (x, 0) = v0ε (x), wε (x, 0) = v0ε (x), x ∈ Ω.

Lemma 3.7 ([42]). Let Ω be a bounded domain in RN (N ≥ 3) with smooth boundary. Assume there is a constant C > 0 such that ∥u∥Ls (Ω) ≤ C f or all t ∈ (0, T ). If v0 ∈ W 1,∞ (Ω ), then there exists some constant Cq such that for every t ∈ (0, T ) and 1 ≤ s < N , the solution of the problem ∂v vt = △v − α1 v + β1 u in Ω , = 0 on ∂Ω ∂ν satisfies ∥v(t)∥W 1,q (Ω) ≤ Cq (3.58) for all q <

Ns N −s .

If s = N , then (3.58) holds for all q < ∞, and if s > N , (3.58) holds for q = ∞.

570

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

Lemma 3.8. There exists ρ0 > 0 such that inf v(x, t) ≥ ρ0 > 0

f or all t ≥ 0,

x∈Ω

where ρ0 does not depend on t. Proof . The proof is similar to [43], so we omitted it. Lemma 3.9 ([18]). Let Ω be a bounded domain in RN (N ≥ 3) with smooth boundary. For any ε > 0 and p > 1, there exists a constant C > 0 such that for each u ∈ L1 (Ω ), the solution of the problem −△w + α2 w = β2 u satisfies

∫

p

∫

w ≤ε Ω

∂w =0 ∂ν

in Ω ,

on ∂Ω

up + C.

(3.59)

Ω

Lemma 3.10. Let Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary, assume that (uε , vε , wε ) is a classical solution to the system (3.57) on [0, Tmax ). Suppose Ψ (uε ), φ(vε ), ψ(wε ) and f (uε ) satisfy (1.6), (1.7) and (1.8), D(uε ) satisfies (1.4) and (1.5). Then with ϵ ≥ β + 1, β > 1, for all ε ∈ (0, 1), we have ∫ 2 |∇vε | ≤ C (3.60) Ω

where C > 0 is a constant independent of t. Proof . Multiplying the first equation of (3.57) by ln uε and integrating by parts yield that ∫ ∫ d uε ln uε = ∇(Dε (uε )∇uε − χuε ∇φ(vε ) + Ψε (uε )∇ψ(wε )) ln uε dt Ω Ω∫ ∫ + f (uε ) ln uε + f (uε ) Ω Ω ∫ ∫ ∫ m+1 2 aχ 4CD 2 | + ∇u ∇v − µ uϵε |∇u ≤− ε ε ε (m + 1)2 Ω ρk01 Ω Ω ∫ ∫ bξ − k uβ−1 ∇u ∇w + ϱ ln uε + ϱ|Ω |, ε ε ε ρ0 2 Ω Ω

(3.61)

where we have used the assumptions (1.4)–(1.7), Lemma 2.2, Lemma 3.7. We multiply the second equation of (3.57) by −△vε and integrating by parts yield that ∫ ∫ ∫ ∫ 1 d 2 2 2 |∇vε | + |△vε | + α1 |∇vε | = β1 ∇uε ∇vε . (3.62) 2 dt Ω Ω Ω Ω Since ln uε ≤ uε for all uε > 0, yield that ∫

∫ ln uε ≤ ϱ

ϱ Ω

uε .

(3.63)

Ω

Furthermore, using the third equation of (3.57) and boundary conditions, we can get that ∫ ∫ aχ aχ (β1 + k ) ∇uε ∇vε = −(β1 + k ) uε △vε ρ0 1 Ω ρ01 Ω ∫ ∫ 1 aχ 2 1 2 2 ≤ (β1 + k ) uε + |△vε | . 2 2 Ω ρ0 1 Ω

(3.64)

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571

Similarly, −

bξ ρk02

∫

uβ−1 ∇uε ∇wε ε

=

Ω

=

∫

bξ

βρk02 Ω ∫ α2 bξ βρk02

uβε △wε uβε wε −

Ω

(3.65) β2 bξ βρk02

∫

uβ+1 . ε

Ω

Combining with (3.61)–(3.65), we have ∫ ∫ ∫ m+1 2 1 4CD 1 d 2 2 2 (uε ln uε + |∇vε | ) + |∇u |△vε | | + ε dt Ω 2 (m + 1)2 Ω 2 Ω ∫ ∫ ∫ β2 bξ 2 β+1 + α1 |∇vε | + k uε + µ uϵε βρ02 Ω Ω Ω ∫ ∫ ∫ 1 aχ 2 α2 bξ 2 β uε + k uε wε + ϱ uε + ϱ|Ω |. ≤ (β1 + k ) 2 ρ0 1 βρ02 Ω Ω Ω

(3.66)

since ϵ ≥ β + 1, by the Young’s inequality, with some positive constants c, we have ∫ ∫ β2 bξ β+1 uε ≤ µ uϵε + c, βρk02 Ω Ω in which (3.66) implies that ∫ ∫ ∫ m+1 2 d 1 4CD 1 2 2 2 (uε ln uε + |∇vε | ) + |∇u | + |△vε | ε dt Ω 2 (m + 1)2 Ω 2 Ω ∫ ∫ 2β2 bξ 2 + α1 |∇vε | + uβ+1 ε βρk02 Ω Ω ∫ ∫ ∫ aχ 1 α2 bξ ≤ (β1 + k )2 uβε wε + ϱ uε + ϱ|Ω |. u2ε + k 2 ρ0 1 βρ02 Ω Ω Ω

(3.67)

By β > 1 and (3.59), using the Young’s inequality, with some positive constants c1 , c2 , we have ∫ ∫ ∫ β2 bξ α2 bξ β β+1 u w ≤ u + c wεβ+1 (3.68) 1 ε ε ε βρk02 Ω 8βρk02 Ω Ω ∫ β2 bξ uβ+1 + c2 , ≤ ε 4βρk02 Ω ∫ ∫ ∫ 2 2 noting the fact 2β Ω uε ln uε ≤ 2β Ω u2ε , which together with 12 (β1 + aχ u , by the Young’s inequality, k1 ) Ω ε µ0

there exists some positive constant c3 1 aχ 2β + (β1 + k )2 2 ρ01 similarly,

∫

uε ≤ Ω

β2 bξ

≤

8βρk02

Ω

∫ ϱ

u2ε

β2 bξ 8βρk02

∫

∫

uβ+1 + c3 , ε

(3.69)

Ω

uβ+1 + c4 ε

(3.70)

Ω

with c4 > 0. The combination of (3.67)–(3.70) yields ∫ ∫ ∫ ∫ ∫ m+1 2 d 1 4CD 1 2 2 2 β2 bξ 2 (uε ln uε + |∇vε | )+ |∇u | + |△v | +α |∇v | + uβ+1 ≤ c5 , (3.71) ε ε 1 ε ε dt Ω 2 (m + 1)2 Ω 2 Ω βρk02 Ω Ω where c5 = c2 + c3 + c4 + ϱ|Ω |. Set y(t) :=

2

∫ Ω

(uε ln uε + 21 |∇vε | ), we have y ′ (t) + 2βy(t) ≤ c5 , which implies

y(t) ≤ y(0) + c6

f or all t ∈ (0, Tmax ),

(3.72)

572

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

where c6 > 0. Using the facts −uε ln uε ≤ e−1 for all uε > 0 and the definition of y(t), then from (3.72), we have ∫ ∫ ∫ ∫ 1 1 2 2 |∇vε | ≤ |∇v0ε | − u0ε ln u0ε + uε ln uε + c6 2 Ω 2 Ω Ω ∫ ∫Ω |Ω | 1 2 |∇v0ε | + + c6 , ≤ u0ε ln u0ε + 2 e Ω Ω the desired results are proved. Lemma 3.11. Let p > 1, q ≥ 2. Then, there exist two positive constants C1 , C2 such that for all t ∈ (0, Tmax ), the solution of (3.57) satisfies d dt

∫ (

upε + |∇vε |

2q

)

Ω

β2 ξp(p − 1)

+

2CD p(p − 1) (p + m)2

∫

p+m 2 2

|∇uε Ω

| +

2(q − 1) q

∫

q 2

|∇|∇vε | | Ω

∫

uεp+β 2(p + β − 1)ρk02 Ω ∫ ∫ ∫ χ2 p(p − 1) 2 2q−2 2 up−m |∇v | + C u |∇v | + p up−1 f (uε ) + C2 . ≤ ε 1 ε ε ε ε 2CD Ω Ω Ω +

Proof . Multiplying the first equation of (3.57) by puεp−1 and integrating by parts yield that ∫ ∫ ∫ d 2 p p−2 u = −p(p − 1) uε Dε (uε )|∇uε | + χp(p − 1) up−1 φ′ (vε )∇uε ∇vε ε dt Ω ε Ω Ω ∫ ∫ ′ − p(p − 1) up−2 Ψ (u )ψ (v )∇u ∇w + p uεp−1 f (uε ) ε ε ε ε ε ε Ω Ω ∫ ∫ χp(p − 1) 2 p+m−2 uεp−1 ∇uε ∇vε ≤ −CD p(p − 1) uε |∇uε | + ρk01 Ω Ω ∫ ∫ ξp(p − 1) p+β−1 + uε (α2 wε − β2 uε ) + p up−1 f (uε ) ε (p + β − 1)ρk02 Ω Ω ∫ ∫ CD p(p − 1) 2 2 uεp+m−2 |∇uε | ≤ −CD p(p − 1) up+m−2 |∇uε | + ε 2 Ω Ω ∫ ∫ α2 ξp(p − 1) χ2 p(p − 1) 2 p−m + uε |∇vε | + uεp+β−1 wε 1 2ρ2k (p + β − 1)ρk02 Ω Ω 0 CD ∫ ∫ β2 ξp(p − 1) p+β uε + p uεp−1 f (uε ), − (p + β − 1)ρk02 Ω Ω which implies ∫ ∫ CD p(p − 1) β2 ξp(p − 1) 2 p+m−2 + uε |∇uε | + uεp+β 2 (p + β − 1)ρk02 Ω Ω Ω ∫ ∫ ∫ χ2 p(p − 1) α2 ξp(p − 1) 2 p−m p+β−1 u |∇v | u w + p uεp−1 f (uε ). ≤ + ε ε ε ε k2 1 2ρ2k C (p + β − 1)ρ Ω Ω Ω D 0 0 d dt

∫

upε

By the Young’s inequality and (3.59), there exist some positive constants c1 , c2 such that ∫ ∫ ∫ α2 ξp(p − 1) β2 ξp(p − 1) p+β−1 p+β uε wε ≤ uε + c1 wεp+β (p + β − 1)ρk02 Ω 4(p + β − 1)ρk02 Ω Ω ∫ β2 ξp(p − 1) ≤ uεp+β + c2 . 2(p + β − 1)ρk02 Ω

(3.73)

(3.74)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

573

Substituting (3.74) into (3.73), we have ∫ ∫ CD p(p − 1) β2 ξp(p − 1) 2 p+m−2 uεp+β uε + |∇uε | + 2 2(p + β − 1)ρk02 Ω Ω Ω ∫ ∫ χ2 p(p − 1) 2 p−m ≤ u |∇v | + p up−1 f (uε ). ε ε ε 1 2ρ2k C Ω Ω D 0 d dt

∫

upε

(3.75)

Differentiating the second equation of system (3.57) and multiplying the results by 2∇vε , we have 2

2

(|∇vε | )t = 2∇vε ∇△vε − 2α1 |∇vε | + 2β1 ∇uε ∇vε , 2

2

2

substituting the fact △|∇vε | = 2∇vε ∇△vε − 2α1 |∇vε | + 2|D2 vε | into above equation, we get 2

2

2

2

(|∇vε | )t = △|∇vε | − 2|D2 vε | − 2α1 |∇vε | + 2β1 ∇uε ∇vε .

(3.76)

2q−2

Multiplying the above equation by q|∇vε | (q ≥ 2), integrating by parts over Ω and using the following estimate [44] ∫ ∫ 2 q(q − 1) 2q−4 2 2 2q−2 ∂|∇vε | dS ≤ |∇vε | |∇|∇vε | | + c3 , q |∇vε | ∂ν 4 Ω ∂Ω with a constant c3 , yield that ∫ ∫ ∫ d 2 2q 2q−4 2 2 2q−2 |∇vε | + q(q − 1) |∇vε | |∇|∇vε | | + 2q |∇vε | |D2 vε | dt Ω Ω Ω ∫ 2q + 2qα1 |∇vε | Ω ∫ ∫ q(q − 1) 2q−4 2 2 2q−2 ≤ |∇vε | |∇|∇vε | | + 2qβ1 |∇vε | ∇uε ∇vε + c4 4 Ω Ω ∫ ∫ q(q − 1) 2q−4 2 2 2q−2 |∇vε | |∇|∇vε | | + 4q(q − 1)β12 (uε + ε)2 |∇vε | ≤ 2 Ω Ω ∫ ∫ 2q qN β1 2q−2 2 2q−2 2 + |∇vε | |△vε | + (uε + ε) |∇vε | + c4 N Ω 2 Ω 2

with a constant c4 , the Young’s inequality we have been used, substituting the inequality |△vε | ≤ N |D2 vε | into the above inequality, we have d dt

∫

2q

|∇vε | Ω

q(q − 1) + 2

∫

2q−4

|∇vε | Ω

N |∇|∇vε | | ≤ qβ1 ( + 4q − 4) 2 2 2

∫

2q−2

(uε + ε)2 |∇vε |

+ c4 .

2

(3.77)

Ω

Combining with (3.75) and (3.77), the desired results are obtained. Lemma 3.12. Let ϵ ≥ β + 1, β + 2m ≥ 2 and N ≥ 3. Then for all β + 2m < p, there exists a constant C > 0 independent of t such that the solution of (3.57) satisfies ∥uε (·, t)∥Lp (Ω) ≤ C

f or all

t ∈ (0, Tmax ),

and ∥∇vε (·, t)∥

2(p+β)

L β+m (Ω)

≤C

f or all

t ∈ (0, Tmax ).

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

574

p+β Proof . For 2 ≤ β + 2m < p and q = β+m > 2, from Lemma 3.10, there exist two positive constants c1 , c2 such that ∫ ( ∫ ) 2C p(p − 1) ∫ p+m 2 2(q − 1) d D 2q q 2 2 upε + |∇vε | + |∇u | + |∇|∇vε | | ε dt Ω (p + m)2 q Ω Ω ∫ β2 ξp(p − 1) p+β + uε (3.78) 2(p + β − 1)ρk02 Ω ∫ ∫ ∫ 2 2q−2 2 ≤ c1 up−m |∇v | + c u |∇v | + p up−1 f (uε ) + c2 . ε 1 ε ε ε ε Ω

Ω

Ω

By the H¨ older inequality, we can choose q = ∫

p+β β+m

2

up−m |∇vε | ≤ ε

> 2 and γ =

(∫

Ω

uεp+β

) p−m (∫ p+β

2q−2

u2ε |∇vε |

≤

(∫

Ω

> 1 such that 2q

|∇vε |

) 1q

(3.79)

Ω

Ω

and ∫

p+β p+β−2

uεp+β

)

2 p+β

(∫

|∇vε |

2(q−1)γ

) γ1

.

(3.80)

Ω

Ω

Furthermore, by the Gagliardo–Nirenberg inequality, there exists a positive constant c3 such that )1 (∫ ( ) 2q 2q q q q q 1 |∇vε | ≤ c3 ∥∇|∇vε | ∥σL12 (Ω) ∥|∇vε | ∥1−σ + c3 ∥|∇vε | ∥ 2 2 where σ1 =

N (q−1) qN +2−N

L q (Ω)

L q (Ω)

Ω

(3.81)

∈ (0, 1). From (3.60) and (3.81), we have (∫

|∇vε |

2q

) 1q

≤ c4

(∫

q 2

|∇|∇vε | |

) σq1

+ c4 .

(3.82)

Ω

Ω

Similarly, there exists a positive constant c5 such that (∫

|∇vε |

2(q−1)γ

) γ1

≤ c5

(∫

Ω

where σ2 =

N (pq−p−m+2) p(qN +2−N ) .

q 2

|∇|∇vε | |

) σ2 (q−1) q

+ c5

(3.83)

Ω

Combining with (3.79), (3.80), (3.82) and (3.83), we have

∫

∫ 2 2q−2 up−m |∇v | + c u2ε |∇vε | ε 1 ε Ω Ω (∫ ) p−m (∫ ) σ1 (∫ ) p−m p+β q 2 q p+β p+β ≤ c1 c4 uε |∇|∇vε | | + c1 c4 uεp+β c1

Ω

Ω

+ c1 c5

(∫

up+β ε

)

2 p+β

Ω

(3.84)

Ω

(∫

q 2

|∇|∇vε | |

) σ2 (q−1) q

+ c1 c5

Ω

(∫

up+β ε

)

2 p+β

.

Ω

σ2 (q−1) σ1 p+β 2 > 2, it is easy to know that p−m < 1. Since q = β+m p+β + q < 1 and if β + 2m ≥ 2, it holds that p+β + q a0 a1 Using the facts x y ≤ ε(x + y) + c6 , where a0 , a1 , c6 > 0 are constants such that a0 + a1 < 1, (3.84) implies that ∫ ∫ ∫ (∫ ) 2 2q−2 q 2 p−m 2 p+β c1 uε |∇vε | + c1 uε |∇vε | ≤ε uε + |∇|∇vε | | + c7 . (3.85) Ω

Ω

Ω

Ω

By (1.7), we have ∫ p Ω

up−1 f (uε ) ε

∫ ≤ pϱ Ω

uεp−1

∫ − µp Ω

up+ϵ−1 , ε

(3.86)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

575

(3.78) implies that ∫ ∫ p+m 2CD p(p − 1) 2(q − 1) q 2 2 2 |∇u | |∇|∇vε | | + ε (p + m)2 q Ω Ω Ω ∫ ∫ β2 ξp(p − 1) p+β + uε + µp up+ϵ−1 ε 2(p + β − 1)ρk02 Ω Ω ∫ ∫ ∫ 2 2q−2 p−m 2 ≤ c1 uε |∇vε | + c1 uε |∇vε | + pϱ uεp−1 + c2 . d dt

∫ (

2q

upε + |∇vε |

)

+

Ω

Ω

Ω

By the Young’s inequality, there exist c8 , c9 , c10 > 0, we have ∫ ∫ pϱ up−1 ≤ c up+β + c9 . 8 ε ε Ω

and

(3.88)

Ω

β2 ξp(p − 1) 2(p + β − 1)ρk02

∫

up+β ε

∫ ≤ µp

Ω

up+ϵ−1 + c10 ε

(3.89)

Ω

Using the Gagliardo–Nirenberg inequality, there exist c11 > 0, we have ∫ 2q q q 2(1−σ ) q 3 |∇vε | ≤ c11 ∥∇|∇vε | ∥2σ ∥|∇vε | ∥ 2 3 + c11 ∥|∇vε | ∥2 2 L2 (Ω) L q (Ω)

Ω

where σ3 =

N (q−1) qN +2−N

(3.87)

∈ (0, 1). From (3.60) and (3.90), we have ∫ (q − 1) q 2q ∥∇|∇vε | ∥2L2 (Ω) + c12 . |∇vε | ≤ q Ω

Moreover, by the H¨ older inequality and Young’s inequality, we have ∫ ( β ξp(p − 1) )∫ 2 upε ≤ + c up+β . 8 ε 4(p + β − 1)ρk02 Ω Ω

(3.90)

L q (Ω)

(3.91)

(3.92)

Substituting (3.85)–(3.87), (3.89), (3.91) and (3.92) into (3.87) implies that ∫ ( ∫ ) ∫ d 2q 2q p p u + |∇vε | + uε + |∇vε | ≤ c13 dt Ω ε Ω Ω there exists a constant c14 such that ∫

upε +

Ω

∫

2q

|∇vε |

≤ c14 .

Ω

Then the desired results are obtained. Lemma 3.13. Let the assumptions in Lemma 3.12 hold. Then there exists C > 0 independent of ε such that ∥uε (·, t)∥L∞ (Ω) ≤ C f or all t ∈ (0, Tmax ), (3.93) ∥vε (·, t)∥W 1,∞ (Ω) ≤ C

f or all

t ∈ (0, Tmax ).

(3.94)

Proof . The proof is similar to Lemma 3.6, so we omitted it. Finally, we prove the main theorem. Proof of Theorem 2.2. Using Lemmas 3.12 and 3.13, the steps are similar to the proof of Theorem 2.1, so we omitted it.

576

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

4. Proof of Theorem 2.3. In this section, we establish some a priori estimates for solutions to the approximated system (2.25) with non-degenerate diffusion, it is crucial ingredient for the proof of our main results. As a first step towards this, we have the following simple estimate. Lemma 4.1. Let f2 ≡ 0, w ≡ 0, τ1 = 1, Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary. Assume that Φε (uε ), φ(vε ) ≡ vε , g(uε ) and f (uε ) satisfy (1.6), (1.8) and (1.9). Suppose that Dε (uε ) satisfies (1.4), and (1.5). Then with ϵ ≥ 2, for all ε ∈ (0, 1), we have ∫ 2 |∇vε | ≤ C (4.1) Ω

where C > 0 is a constant independent of t. Proof . Multiplying the first equation of (2.25) by ln uε and integrating by parts yield that ∫ ∫ ∫ ∫ d uε ln uε = ∇(Dε (uε )∇uε − χuε ∇φ(vε )) ln uε + f (uε ) ln uε + f (uε ) dt Ω Ω Ω Ω ∫ ∫ ∫ m+1 4CD 2 ≤− |∇u |2 + χ ∇uε ∇vε − µ uϵε ε (m + 1)2 Ω Ω Ω ∫ +ϱ ln uε + ϱ|Ω |,

(4.2)

Ω

where we have used the assumptions (1.5)–(1.8), Lemmas 3.7 and 3.8. Let Cg := ∥g∥L∞ ((0,∥v0 ∥

W 1,∞ (Ω)

+1)) .

In fact, according to (1.9), we have g(vε ) ≤ ∥g∥L∞ ((0,∥v0ε ∥L∞ (Ω) +1)) ≤ ∥g∥L∞ ((0,∥v0 ∥

W 1,∞ (Ω)

+1))

= Cg

We multiply the second equation of (1.7) by −△vε and integrating by parts yield that ∫ ∫ ∫ 1 d 2 2 |∇vε | + |△vε | = uε △vε g(vε ) 2 dt Ω Ω Ω∫ ∫ 1 2 ≤ |△vε | + Cg2 u2ε 4 Ω Ω

(4.3)

Since ln uε ≤ uε for all uε > 0, yield that ∫

∫ ln uε ≤ ϱ

ϱ Ω

uε .

Furthermore, using the third equation of (3.57) and boundary conditions, we can get that ∫ ∫ ∫ ∫ 1 2 2 2 χ ∇uε ∇vε = −χ uε △vε ≤ χ uε + |△vε | . 4 Ω Ω Ω Ω Combining with (4.2)–(4.5), we have ∫ ∫ ∫ ∫ m+1 1 4CD 1 d 2 2 2 2 (uε ln uε + |∇vε | ) + |∇uε | + |△vε | + µ uϵε dt Ω 2 (m + 1)2 Ω 2 Ω Ω ∫ ∫ ≤ (a2 χ2 + Cg2 ) u2ε + ϱ uε + ϱ|Ω |. Ω

Ω

(4.4)

Ω

(4.5)

(4.6)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

577

∫ ∫ ∫ Noting the fact 2β Ω uε ln uε ≤ 2β Ω u2ε , which together with (a2 χ2 +Cg2 ) Ω u2ε , since ϵ ≥ 2, by the Young’s inequality, with some positive constants c1 , c2 , we have ∫ ∫ 1 uϵε + c1 , (4.7) (a2 χ2 + Cg2 + 2β) u2ε ≤ 2µ Ω Ω and

∫ uε ≤

ϱ Ω

1 2µ

∫

uϵε + c2 ,

(4.8)

Ω

in which (4.6) implies that ∫ ∫ ∫ ∫ m+1 d 1 4CD 1 2 2 2 2 (uε ln uε + |∇vε | ) + |∇uε | + |△vε | + 2β uε ln uε ≤ c3 , dt Ω 2 (m + 1)2 Ω 2 Ω Ω

(4.9)

where c3 = c1 + c2 + ϱ|Ω |, using the Young’s inequality, we have ∫ ∫ ∫ 1 2 2 |△vε | + vε2 |∇vε | ≤ 4 Ω Ω Ω ∫ 2 Set y(t) := Ω (uε ln uε + 12 |∇vε | ), we have y ′ (t) + 2βy(t) ≤ c4 , which implies y(t) ≤ y(0) + c5

f or all t ∈ (0, Tmax ),

(4.10)

where c5 > 0. Using the facts −uε ln uε ≤ e−1 for all uε > 0 and the definition of y(t), then from (3.72), we have ∫ ∫ ∫ ∫ 1 1 2 2 |∇vε | ≤ u0ε ln u0ε + |∇v0ε | − uε ln uε + c5 2 Ω 2 Ω Ω Ω ∫ ∫ 1 |Ω | 2 ≤ u0ε ln u0ε + |∇v0ε | + + c5 , 2 Ω e Ω the desired results are proved. Lemma 4.2. Let p > 1, q ≥ 2. Then there exist two positive constants C1 , C2 such that for all t ∈ (0, Tmax ), the solution of (2.25) satisfies ∫ ( ∫ ) 2C p(p − 1) ∫ p+m 2(q − 1) d D 2q q 2 2 2 upε + |∇vε | + |∇|∇vε | | |∇u | + ε dt Ω (p + m)2 q Ω Ω ∫ ∫ ∫ χ2 p(p − 1) 2 2q−2 p−m 2 ≤ uε |∇vε | + C1 uε |∇vε | +p uεp−1 f (uε ) + C2 . 2CD Ω Ω Ω Proof . Multiplying the first equation of (2.25) by puεp−1 and integrating by parts yields that ∫ ∫ ∫ d 2 upε = −p(p − 1) up−2 D (u )|∇u | + χp(p − 1) uεp−1 φ′ (vε )∇uε ∇vε ε ε ε ε dt Ω Ω Ω ∫ +p up−1 f (uε ) ε Ω ∫ ∫ ∫ 2 ≤ −CD p(p − 1) uεp+m−2 |∇uε | + χp(p − 1) uεp−1 ∇uε ∇vε + p up−1 f (uε ) ε Ω Ω Ω ∫ ∫ CD p(p − 1) 2 2 up+m−2 |∇uε | ≤ −CD p(p − 1) up+m−2 |∇uε | + ε ε 2 Ω ∫Ω ∫ χ2 p(p − 1) 2 p−m p−1 + uε |∇vε | + p uε f (uε ), 2CD Ω Ω

578

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

which implies ∫ ∫ ∫ ∫ d CD p(p − 1) χ2 p(p − 1) 2 2 p p+m−2 p−m u + uε uε |∇vε | + p |∇uε | ≤ uεp−1 f (uε ). dt Ω ε 2 2CD Ω Ω Ω By calculation to the second equation in (2.3), we obtain ∫ ∫ d 2q 2q−2 |∇vε | = 2q |∇vε | ∇vε · ∇vεt dt Ω ∫Ω 2q−2 = 2q |∇vε | ∇vε · ∇(△vε − uε g(vε )) ∫Ω ∫ 2q−2 2q−2 = 2q |∇vε | ∇vε · ∇△vε − 2q |∇vε | ∇vε · ∇(uε g(vε )) Ω Ω ∫ ∫ 2 2q−2 2 2q−2 1 |∇vε | ∇vε · ∇(uε g(vε )) |∇vε | ( △|∇vε | − |D2 vε | ) − 2q = 2q 2 Ω Ω ∫ ∫ 2 2q−2 ∂|∇vε | 2q−4 2 2 =q |∇vε | − q(q − 1) |∇vε | |∇|∇vε | | ∂ν ∂Ω∫ Ω ∫ 2 2q−2 2q−2 2 − 2q |∇vε | |D vε | − 2q |∇vε | ∇vε · ∇(uε g(vε )), Ω

(4.11)

(4.12)

Ω 2

2

where using the pointwise identity ∇vε · ∇△vε = 12 △|∇vε | − |D2 vε | . In view of Lemma 2.2–2.5 in [44], we can find some positive constants c1 such that ∫ ∫ 2 2 2q−2 ∂|∇vε | 2q−2 q |∇vε | ≤q |∇vε | |D2 vε | + c1 . (4.13) ∂ν ∂Ω Ω Integrating by parts for the last part of (3.6), we get ∫ ∫ 2q−2 2q−2 − 2q |∇vε | ∇vε · ∇(uε f (vε )) = 2q uε f (vε )∇ · (|∇vε | ∇vε ) Ω ∫Ω 2q−2 = 2q uε f (vε )|∇vε | △vε Ω ∫ 2q−4 2 + 2q(q − 1) uε f (vε )|∇vε | ∇|∇vε | · ∇vε

(4.14)

Ω

=: J1 + J2 . 2

2

Using Young’s inequality and |△vε | ≤ N |D2 vε | , we arrive at the inequality ∫ ∫ q 2q−2 2 2q−2 J1 ≤ |∇vε | |△vε | + qN (uε f (vε ))2 |∇vε | N Ω Ω ∫ ∫ 2 2q−2 2q−2 ≤q |∇vε | |D2 vε | + qN Cg2 u2ε |∇vε | Ω

(4.15)

Ω

and ∫ q(q − 1) 2q−4 2 2q−4 2 2 (uε f (vε ))2 |∇vε | |∇vε | + |∇vε | |∇|∇vε | | 2 Ω Ω ∫ ∫ q(q − 1) 2q−2 2q−4 2 2 ≤ 2q(q − 1)Cg2 u2ε |∇vε | + |∇vε | |∇|∇vε | | . 2 Ω Ω ∫

J2 ≤ 2q(q − 1)

Inserting (4.13)–(4.16) into (4.12), we obtain ∫ ∫ ∫ d q(q − 1) N 2q 2q−4 2 2 2q−2 |∇vε | + |∇vε | |∇|∇vε | | ≤ 2q(q − 1 + )Cg2 u2ε |∇vε | + c1 , dt Ω 2 2 Ω Ω Combining with (4.11) and (4.17), the desired results are obtained.

(4.16)

(4.17)

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

579

Lemma 4.3. Let ϵ ≥ 2, m ≥ 1 − ϵ and N ≥ 3. Then for all 2m + ϵ − 1 < p, there exists a constant C > 0 independent of t such that the solution of (2.17) satisfies ∥uε (·, t)∥Lp (Ω) ≤ C

t ∈ (0, Tmax ),

f or all

and ∥∇vε (·, t)∥

≤C

2(p+ϵ−1)

t ∈ (0, Tmax ).

f or all

L m+ϵ−1 (Ω) p+ϵ−1 > 2, from Lemma 4.2, there exist two positive Proof . For 2m + 1 ≤ 2m + ϵ − 1 < p and q = m+ϵ−1 constants c1 , c2 such that ∫ ( ∫ ) 2C p(p − 1) ∫ p+m d 2(q − 1) D 2q q 2 2 2 | + upε + |∇vε | + |∇u |∇|∇vε | | (4.18) ε dt Ω (p + m)2 q Ω Ω ∫ ∫ ∫ 2 2q−2 ≤ c1 up−m |∇vε | + c1 u2ε |∇vε | +p up−1 f (uε ) + c2 . ε ε Ω

Ω

Ω

By the H¨ older inequality, we can choose q = ∫

p+ϵ−1 m+ϵ−1

(∫

2

up−m |∇vε | ≤ ε

∫

up+ϵ−1 ε

p+ϵ−1 p+ϵ−3

p−m (∫ ) p+ϵ−1

Ω

Ω

and

> 2 and η =

2q−2 u2ε |∇vε |

≤

(∫

2q

|∇vε |

) 1q

(4.19)

Ω

uεp+ϵ−1

2 ) p+ϵ−1 (∫

Ω

Ω

> 1 such that

2(q−1)η

|∇vε |

) η1

.

(4.20)

Ω

Furthermore, by the Gagliardo–Nirenberg inequality, there exists a positive constant c3 such that )1 ( (∫ ) 2q 2q q q q q 1 |∇vε | ≤ c3 ∥∇|∇vε | ∥σL12 (Ω) ∥|∇vε | ∥1−σ + c ∥|∇v | ∥ 2 3 ε 2 where τ1 =

N (q−1) qN +2−N

L q (Ω)

L q (Ω)

Ω

∈ (0, 1). From (4.1) and (4.21), we have (∫

2q

|∇vε |

) 1q

≤ c4

Ω

(∫

q 2

|∇|∇vε | |

) σq1

+ c4 .

(4.22)

Ω

Similarly, there exists a positive constant c5 such that (∫ )1 (∫ ) τ2 (q−1) q 2(q−1)η η q 2 ≤ c5 + c5 |∇vε | |∇|∇vε | | Ω

where τ2 =

(4.21)

(4.23)

Ω

N (pq−pm−p−ϵ+3) (p−m)(qN +2−N ) .

Combining with (4.19), (4.20), (4.22) and (4.23), we have ∫ ∫ 2 2q−2 c1 up−m |∇v | + c u2ε |∇vε | ε 1 ε Ω Ω p−m (∫ p−m (∫ ) p+ϵ−1 ) σ1 (∫ ) p+ϵ−1 q 2 q ≤ c1 c4 uεp+ϵ−1 |∇|∇vε | | + c1 c4 up+ϵ−1 ε Ω

+ c1 c5

Ω

(∫

up+ϵ−1 ε

2 ) p+ϵ−1 (∫

Ω

(4.24)

Ω q 2

|∇|∇vε | |

) σ2 (q−1) q

+ c1 c5

(∫

Ω

uεp+ϵ−1

2 ) p+ϵ−1

.

Ω

p+ϵ−1 p−m Since q = m+ϵ−1 > 2, it is easy to know that p+ϵ−1 + σq1 < 1 and if ϵ ≥ 2, m ≥ 1 − ϵ, it holds that σ2 (q−1) 2 < 1. Using the facts xa0 y a1 ≤ ε(x + y) + c6 , where a0 , a1 , c6 > 0 are constants such that p+ϵ−1 + q a0 + a1 < 1, (4.24) implies that ∫ ∫ ∫ (∫ ) 2 2q−2 q 2 2 p+ϵ−1 c1 up−m |∇v | + c u |∇v | ≤ ε u + |∇|∇vε | | + c7 . (4.25) ε 1 ε ε ε ε Ω

Ω

Ω

Ω

G. Ren, B. Liu / Nonlinear Analysis: Real World Applications 46 (2019) 545–582

580

By (1.9), we have ∫ p

up−1 g(uε ) ε

∫

up−1 ε

≤ pϱ

Ω

∫ − µp

Ω

up+ϵ−1 , ε

(4.26)

Ω

(4.18) implies that ∫ ( ∫ ∫ ) 2C p(p − 1) ∫ p+m 2(q − 1) d D 2q q 2 p 2 2 |∇uε u + |∇vε | + |∇|∇vε | | + µp uεp+ϵ−1 | + dt Ω ε (p + m)2 q Ω Ω Ω ∫ ∫ ∫ 2 2q−2 p−m 2 p−1 ≤ c1 uε |∇vε | + c1 uε |∇vε | + pϱ uε + c2 . (4.27) Ω

Ω

Ω

By the Young’s inequality, there exist c8 > 0, we have ∫ ∫ µp up−1 ≤ up+ϵ−1 + c8 . pϱ ε 2 Ω ε Ω

(4.28)

Using the Gagliardo–Nirenberg inequality, there exist c9 > 0, we have ∫ 2q q q 2(1−σ ) q 3 |∇vε | ≤ c9 ∥∇|∇vε | ∥2σ ∥|∇vε | ∥ 2 3 + c9 ∥|∇vε | ∥2 2 L2 (Ω) L q (Ω)

Ω

where σ3 =

N (q−1) qN +2−N

(4.29)

L q (Ω)

∈ (0, 1). From (4.1) and (4.29), we have ∫

2q

|∇vε |

≤

Ω

(q − 1) q ∥∇|∇vε | ∥2L2 (Ω) + c10 . q

Moreover, by the H¨ older inequality and Young’s inequality, we have ∫ ∫ µp up+ϵ−1 + c11 . upε ≤ 2 Ω ε Ω

(4.30)

(4.31)

Substituting (4.25), (4.28)–(4.31) into (4.27) implies that ∫ ( ∫ ) ∫ d 2q 2q upε + |∇vε | + upε + |∇vε | ≤ c13 dt Ω Ω Ω there exists a constant c14 such that ∫

upε

∫

Ω

2q

|∇vε |

+

≤ c14 .

Ω

Then the desired results are obtained. Lemma 4.4. Let the assumptions in Lemma 4.3 hold. Then there exists C > 0 independent of ε such that ∥uε (·, t)∥L∞ (Ω) ≤ C ∥vε (·, t)∥W 1,∞ (Ω) ≤ C

f or all f or all

t ∈ (0, Tmax ), t ∈ (0, Tmax ).

(4.32) (4.33)

Proof . The proof is similar to Lemma 3.6, so we omitted it. Finally, we prove the main theorem. Proof of Theorem 2.3. Using Lemmas 4.3 and 4.4, the steps are similar to the proof of Theorem 2.1, so we omitted it.

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581

5. Concluding remark In this paper, we mainly investigated the global bounded classical and weak solutions to the system (1.3), our results in Theorem 2.1(1) generalize earlier results in [35, Theorem 1(i)] and [19, Theorem 1.2(ii)], Theorem 2.2 generalized earlier results in [10, Theorem 1.1(ii)], Corollary 2.1 is consistent with the results in [36, Theorem 1.3], and improved the results in [21, Theorem 1.1(ii)]. We notice that in Theorem 2.1, it is unclear the existence of global bounded weak solution of (1.3) in the coexistence case when f1 ̸= 0 and f2 ̸= 0, moreover, it is not clear that the fact mentioned above even if f1 ≡ 0, k1 < 1, k2 > 1 or k1 > 1, k2 < 1, we have to discuss those two problems in our forthcoming paper. Acknowledgments The authors express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript. References [1] P.F. Verhulst, Notice sur la loique la population poursuitdans son accroissement, Correspondance Math. Phys. 10 (1838) 113–121. [2] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415. [3] T. Hillen, K.J. Painter, A user’s guide to P DE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217. [4] I. Tuval, L. Cisneros, C. Dombrowski, C.W. Wolgemuth, J.O. Kessler, R.E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. 102 (2005) 2277–2282. [5] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations 37 (2012) 319–352. [6] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ration. Mech. Anal. 211 (2014) 455–487. [7] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015) 3789–3828. [8] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Am. Math. Soc. 369 (5) (2017) 3067–3125. [9] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet, A. Mogilner, Chemotactic signalling, microglia, and Alzheimer’s disease senile plague: is there a connection? Bull. Math. Biol. 65 (2003) 673–730. [10] H.Y. Jin, Z.A. Wang, A dual-gradient chemotaxis system modeling the spontaneous aggregation of microglia in Alzheimer’s disease, Anal. Appl. (2017) 1–32. [11] S. Wu, B. Wu, Global boundedness in a quasilinear attraction–repulsion chemotaxis model with nonlinear sensitivity, J. Math. Anal. Appl. 442 (2) (2016) 554–582. [12] D. Li, C.L. Mu, K. Lin, L.C. Wang, Global weak solutions for an attraction–repulsion system with nonlinear diffusion, Math. Methods Appl. Sci. 40 (18) (2017) 7368–7395. [13] Y.S. Tao, Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci. 23 (1) (2013) 1–36. [14] H.Y. Jin, Boundedness of the attraction–repulsion Keller–Segel system, J. Math. Anal. Appl. 422 (2) (2015) 1463–1478. [15] D.M. Liu, Y.S. Tao, Global boundedness in a fully parabolic attraction–repulsion chemotaxis model, Math. Methods Appl. Sci. 38 (12) (2015) 2537–2546. [16] H.Y. Jin, Z.R. Liu, Large time behavior of the full attraction–repulsion Keller–Segel system in the whole space, Appl. Math. Lett. 47 (2015) 13–20. [17] K. Lin, C.L. Mu, Global existence and convergence to steady states for an attraction–repulsion chemotaxis system, Nonlinear Anal. RWA 31 (2016) 630–642. [18] Q.S. Zhang, Y.X. Li, An attraction–repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech. 96 (5) (2016) 570–584. [19] Y.L. Wang, A quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source, J. Math. Anal. Appl. 441 (2016) 259–292. [20] W. Wang, M.D. Zhuang, S.N. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction–repulsion chemotaxis system with logistic source, J. Differential Equations 264 (2018) 2011–2027. [21] P. Zheng, C. Mu, X. Hu, Boundedness in the higher dimensional attraction–repulsion chemotaxis-growth system, Comput. Math. Appl. 72 (2016) 2194–2202. [22] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl. 381 (2011) 521–529.

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