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Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source
Computers and Mathematics with Applications 72 (2016) 2604–2619
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Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source Jiashan Zheng a,∗ , Yifu Wang b a
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, PR China
b
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China
article
info
Article history: Received 11 March 2016 Received in revised form 5 August 2016 Accepted 17 September 2016 Available online 17 October 2016 Keywords: Boundedness Chemotaxis Global existence Decay behavior Logistic source
abstract This paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype
ut = ∆um − ∇ · (u∇v) + µu − ur , x ∈ , t > 0, vt = ∆v − v + u, x ∈ , t > 0, ∂v ∂u = = 0, x ∈ ∂ , t > 0, ∂ν ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x) x ∈ ,
(0.1)
in a smooth bounded domain ⊂ RN (N ≥ 2) with parameters m, r ≥ 1 and µ ≥ 0. The PDE system in (0.1) is used in mathematical biology to model the mechanism of chemotaxis, that is, the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space. It is shown that if
N +2 2 if 1 < r < , >2− N N + N +2 N +2 (N + 2 − 2r ) m >1+ if ≥r≥ , N +2 2 N ≥ 1 if r > N + 2 , 2
¯ ) × W 1,∞ ()(ι > 0), then (0.1) possesses and the nonnegative initial data (u0 , v0 ) ∈ C ι ( at least one global bounded weak solution. Apart from this, it is proved that if µ = 0 then both u(·, t ) and v(·, t ) decay to zero with respect to the norm in L∞ () as t → ∞. © 2016 Elsevier Ltd. All rights reserved.
∗
Corresponding author. E-mail address: [email protected] (J. Zheng).
http://dx.doi.org/10.1016/j.camwa.2016.09.020 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
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1. Introduction In this paper, we consider the Neumann initial–boundary value problem for a full chemotaxis system with generalized volume-filling effect and nonlinear logistic source
ut = ∇ · (D(u)∇ u) − χ∇ · (S (u)∇v) + µu − ur , vt = 1v + u − v, x ∈ Ω , t > 0, ∂u ∂v = = 0, x ∈ ∂ Ω , t > 0, ∂ν ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω
x ∈ Ω , t > 0, (1.1)
in a bounded domain Ω ⊂ RN (N ≥ 2) with smooth outward normal vector field ν on ∂ Ω , where S (u) = u is the sensitivity function, parameter χ > 0, and the initial data u0 ∈ C ι (Ω )(ι > 0) and v0 ∈ W 1,∞ (Ω ) are assumed to satisfy u0 , v0 ≥ 0 in Ω. The problem of this type is used to describe theoretically the process of chemotaxis, the biological phenomenon of the oriented movement of cells or organisms in response to a chemical signal. The pioneering works of chemotaxis model were introduced by Patlak [1] in 1953 and Keller and Segel [2] in 1970, and we refer the reader to the survey by Hillen and Painter [3] and Horstmann [4], where a comprehensive information of further examples illustrating the outstanding biological relevance of chemotaxis can be found. In this paper, we assume that the diffusion function D(u) satisfies θ D ∈ Cloc ([0, ∞)) for some θ > 0,
(1.2)
and there exist some constants m ≥ 1 and CD > 0 such that D(u) ≥ CD um−1
for all u ≥ 0.
(1.3)
During the past decades, the variants of (1.1) without cell kinetics have been studied extensively by researchers and the main issue of the investigation was whether the solutions of the models are bounded or blow-up (see e.g. Horstmann and Winkler [5], Tao and Winkler [6], Ishida et al. [7], Winkler [8]). In many applications, blow-up phenomena do not appropriately reflect the respective experimentally observable behavior. Accordingly, considerable efforts have been devoted to developing models in which blow-up phenomena are prevented. In this context it is recognized that cell proliferation terms of logistic type, such as contained in (1.1), form the possibly simplest among the blow-up preventing mechanisms. In fact, the presence of such logistic terms is sufficient to suppress any blow-up in many relevant situations. For example, if r = 2 (quadratic type), Winkler [9] discussed the global boundedness of classical solutions to problem (1.1) on a smooth bounded convex domain under the assumption that either N ≤ 2, or the logistic damping effect is large enough. A large number of facts indicate that Keller–Segel-growth systems with logistic-type kinetics of superlinear, not necessarily quadratic type. In fact, when D(u) = (u + 1)−α , the sensitivity function S (u) = u is replaced by S (u) = (u + 1)β−1 with 0 < α + β < N2 and the logistic term f satisfies f (u) ≤ µ − bur
for all u ≥ 0
(1.4)
with some µ ≥ 0, b > 0 and r = 2, Wang et al. [10] obtained the unique global uniformly bounded classical solution (u, v) of problem (1.1), but there is not any available result on the boundedness of the solution when α + β > N2 . Furthermore, assuming that the logistic source f ∈ C ∞ ([0, ∞)) satisfies f (u) ≤ µu − bu2
for all u ≥ 0
(1.5)
and the diffusion function D and the sensitivity function S fulfill D, S ∈ C 2 ([0, ∞)) and
D(s) ≥ 0 for all s ≥ 0,
c1 s ≤ D(s) for all s ≥ s0 , p
c1 sβ ≤ S (s) ≤ c2 sβ
for all s ≥ s0
with c2 > c1 > 0, s0 > 1 and p, β ∈ R, Cao [11] proved that if β < 1, then the classical solution of (1.1) is globally bounded, whereas the question whether or not blow-up may occur is left therein when β > 1. Recently, in [12], it is proved that if 0 < α + β < max{r − 1 + α, N2 } or b is big enough when β = r − 1, then the classical solutions to the corresponding system are uniformly bounded. There is also other alternative as the degenerate sensitivity function, degenerate problems have been studied to prevent blow up (see Chamoun et al. [13], Lorz [14]). Going beyond these boundedness statements, a number of results are available which show that the interplay of chemotactic cross-diffusion and cell kinetics of logistic-type may lead to quite a colorful dynamics. For instance, if D(u) ≡ 1, S (u) = u, f (u) = u − bu2 and the ratio χb is sufficiently
large, Winkler [15] proved that the unique nontrivial spatially homogeneous equilibrium given by u = v ≡ 1b is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data (u0 , v0 ) such that u0 ̸≡ 0, the problem (1.1) possesses a uniquely determined global classical solution (u, v) with (u, v)|t =0 = (u0 , v0 ) which satisfies
u(·, t ) −
b ∞
1
L
→ 0 and (Ω )
v(·, t ) −
b ∞
1
L
→0 (Ω )
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as t → ∞. When N = 1, D(u) ≡ 1, S (u) = u, f (u) = µu − bu2 with b ≥ 1, Winkler [16] obtained that the solutions of parabolic–elliptic system (the second equation in (1.1) is replaced by an elliptic equation, i.e., 0 = 1v − v + u) may become large at intermediate time scales, thus exceeding the system’s carrying capacity to an arbitrary extent (though not blowing up). If f (u) = µu − buλ (λ > 1), µ, b ≥ 0 and D(u) ≡ 1, S (u) = u, N ≥ 5 and λ < 32 + 2N1−2 , Winkler [17] showed that there exist initial data such that the smooth local-in-time solution of parabolic–elliptic system blows up in finite time. In this paper accounting for the damping effect of logistic term more efficiently, we prove that if
N +2 2 if 1 < r < , >2− N N (N + 2 − 2r )+ N +2 N +2 m >1+ if ≥r≥ , N +2 2 N ≥ 1 if r > N + 2 ,
(1.6)
2 then for proper regularity hypotheses on the initial data, (1.1) possesses at least one globally bounded weak solution. Moreover, if µ = 0, then both u(·, t ) and v(·, t ) decay to zero with respect to the norm in L∞ (Ω ) as t → ∞. Our result quantitatively shows that when the effects of nonlinear diffusivity and logistic kinetics of cells are taken into account simultaneously, the range of relevant parameters can be enlarged, which can ensure the boundedness of the cell density. 4 Indeed, in the case β = 1, r = 2, our result implies that m > 2 − N + can ensure the existence of the bounded solutions, 2
whereas it is m > 2 − N2 in [10,12]. It is worth to remark the main idea underlying the proof of our results. Involving the variation-of-constants formula for v , we have
v(t + τ ) = e−τ (A+Id ) v(t ) +
t +τ
e−(t +τ −s))(A+Id ) u(s)ds,
t ∈ (0, Tmax − τ )
t
for some τ > 0 and the operator A defined below. Then with the help of boundedness of
Ω
T 0
Ω
ur dxdt, we can obtain
|∇v|σ ≤ C for all t ∈ (0, Tmax )
(1.7)
for σ ∈ [1, max{ (N +Nr , N }), and thereby establish the a priori estimates of the functional 2−r )+ N −1
p
2β
u +
|∇v|
Ω
Ω
for any p > 1 and β > 1.
2. Preliminaries and main results Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. To begin with, let us collect some basic solution properties which essentially have already been used in [5]. Lemma 2.1 ([5]). For p ∈ (1, ∞), let A := Ap denote the sectorial operator defined by Ap u := −1u
for all u ∈ D(Ap ) :=
∂ϕ ϕ ∈ W 2,p (Ω ) = 0 . ∂ν ∂ Ω
(2.1)
The fact that the spectrum of A is a p-independent countable set of positive real numbers 0 = µ0 < µ1 < µ2 < · · · entails the following consequences: (i) The operator A + Id possesses fractional powers (A + Id )α (α ≥ 0), the domains of which have the embedding properties D((A + Id )α ) ↩→ W 1,p (Ω )
if α >
1 2
.
(2.2)
(ii) Moreover, for all 1 ≤ p < q < ∞ and u ∈ Lp (Ω ) the general Lp − Lq estimate α −tA
∥(A + Id ) e
u∥Lq (Ω ) ≤ ct
−α− N2
1 1 p−q
e(1−µ)t ∥u∥Lp (Ω ) ,
for any t > 0 and α ≥ 0 with some µ > 0. Lemma 2.2 ([18]). Let 0 < θ ≤ p ≤
2N . N −2
There exists a positive constant CGN such that for all u ∈ W 1,2 (Ω ) ∩ Lθ (Ω ),
∥u∥Lp (Ω ) ≤ CGN (∥∇ u∥aL2 (Ω ) ∥u∥1Lθ−(aΩ ) + ∥u∥Lθ (Ω ) ) N
is valid with a =
N
θ−p 1− N + Nθ 2
∈ (0, 1).
(2.3)
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Lemma 2.3 ([19]). Let s ≥ 1 and q ≥ 1. Assume that p > 0 and a ∈ (0, 1) satisfy 1
−
2
p N
q
= (1 − a) + a s
1 2
−
1
and p ≤ a.
N
s
Then there exist c0 , c0′ > 0 such that for all u ∈ W 1,2 (Ω ) ∩ L q (Ω ),
∥u∥W p,2 (Ω ) ≤ c0 ∥∇ u∥aL2 (Ω ) ∥u∥1−s a
+ c0′ ∥u∥
L q (Ω )
s
L q (Ω )
.
Lemma 2.4 ([20]). Let T > 0, τ ∈ (0, T ), A > 0 and B > 0, and suppose that y : [0, T ) → [0, ∞) is absolutely continuous such that y′ (t ) + Ay(t ) ≤ h(t ) for a.e. t ∈ (0, T ) with some nonnegative function h ∈
L1loc
(2.4)
([0, T )) satisfying
t +τ
h(s)ds ≤ B
for all t ∈ (0, T − τ ).
t
Then
y(t ) ≤ max y0 + B,
B Aτ
+ 2B
for all t ∈ (0, T ).
Theorem 2.1. Let µ ≥ 0. Assume that the initial data (u0 , v0 ) are such that
¯ ) for certain ι > 0 with u0 ≥ 0 in Ω , u0 ∈ C ι ( Ω ¯, v0 ∈ W 1,∞ (Ω ) with v0 ≥ 0 in Ω
(2.5)
D and m satisfy (1.2) – (1.3) and (1.6), respectively. Then the problem (1.1) possesses at least one global weak solution (u, v) in the sense of Definition 2.1. This solution is bounded in Ω × (0, ∞) in the sense that
∥u(·, t )∥L∞ (Ω ) + ∥v(·, t )∥W 1,∞ (Ω ) ≤ C for all t > 0.
(2.6)
¯ × [0, ∞) and Furthermore, v is continuous in Ω 0 u ∈ Cω−∗ ([0, ∞); L∞ (Ω )),
(2.7)
that is, u is continuous on [0, ∞) as an L (Ω )-valued function with respect to the weak-∗ topology. ∞
Theorem 2.2. Let µ = 0, and suppose that D fulfills (1.2)–(1.3) with m satisfying (1.6). Then as long as (u0 , v0 ) fulfills (2.5) with u0 ̸≡ 0, the global weak solution (u, v) constructed in Theorem 2.1 satisfies ∗
∥u(·, t )∥L∞ (Ω ) ⇀ 0,
∥v(·, t )∥L∞ (Ω ) → 0
(2.8)
as t → ∞.
s
Definition 2.1. Let T > 0, (u0 , v0 ) fulfill (2.5) and H (s) = 0 D(σ )dσ for s ≥ 0. Then a pair (u, v) of nonnegative functions is called a weak solution of (1.1) if the following conditions are satisfied
¯ × [0, T )), u ∈ L1loc (Ω
(2.9)
1 1 ,1 ¯ v ∈ L∞ (Ω )), loc (Ω × [0, T )) ∩ Lloc ([0, T ); W
H (u)
¯ × [0, T )), and u|∇v| belong to L1loc (Ω
(2.10)
and T
− Ω
0
uϕt −
Ω
u0 ϕ(·, 0) =
T
H (u)1ϕ + χ
Ω
0
T
Ω
0
u∇v · ∇ϕ + µ
0
T
Ω
ϕu −
0
T
Ω
ϕ ur
(2.11)
∂ϕ
¯ × [0, T )) satisfying = 0 on ∂ Ω × (0, T ) as well as for any ϕ ∈ C0∞ (Ω ∂ν T
− 0
Ω
vϕt −
Ω
v0 ϕ(·, 0) = −
T
T
∇v · ∇ϕ − 0
Ω
0
Ω
vϕ +
T
0
Ω
uϕ
(2.12)
¯ × [0, T )). for any ϕ ∈ C0∞ (Ω If (u, v) is a weak solution of (1.1) in Ω × (0, T ) for all T > 0, then (u, v) is said to be a global weak solution of (1.1).
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
We intend to construct a weak solution of (1.1) as the limit of a sequence of solutions to the regularized systems. Thus, in this section, we firstly establish the regularized system of (1.1). To this end, we approximate the diffusion coefficient function in (1.1) by a family (Dε )ε∈(0,1) of functions Dε ∈ C 2 ((0, ∞))
such that Dε (n) ≥ ε for all n > 0
and D(n) ≤ Dε (n) ≤ D(n) + 2ε
for all n > 0 and ε ∈ (0, 1).
Therefore, the regularized problems of (1.1) are presented as follows
u = ∇ · (D (u )∇ u ) − χ∇ · (u ∇v ) + µu − ur , εt ε ε ε ε ε ε ε vεt = 1vε − vε + uε , x ∈ Ω , t > 0, ∂vε ∂ uε = = 0, x ∈ ∂ Ω , t > 0, ∂ν ∂ν uε (x, 0) = u0 (x), vε (x, 0) = v0 (x), x ∈ Ω .
x ∈ Ω , t > 0, (2.13)
The following local existence result on (2.13) is rather standard, and we like to refer the readers to [21,10,22] for its proof. Lemma 2.5 ([21,10,22]). For any ε ∈ (0, 1), there exist Tmax ∈ (0, ∞] and a classical solution (vε , vε ) of (2.13) in Ω ×(0, Tmax ) such that
¯ × [0, Tmax )) ∩ C 2,1 (Ω ¯ × (0, Tmax )), uε ∈ C 0 ( Ω 0 ¯ ¯ × (0, Tmax )). vε ∈ C (Ω × [0, Tmax )) ∩ C 2,1 (Ω
(2.14)
Moreover, uε and vε are nonnegative in Ω × (0, Tmax ), and
∥uε (·, t )∥L∞ (Ω ) + ∥vε (·, t )∥W 1,∞ (Ω ) → ∞ as t ↗ Tmax .
(2.15)
3. A priori estimates In this section, we are going to establish an iteration step to develop the main ingredient of our result. The iteration depends on a series of a priori estimates. Now, due to the presence of the nonlinear logistic source in the first equation of (1.1), some useful estimates can be derived from Lemma 2.2 in [20], which can be stated in the following. Lemma 3.1. There exists λ > 0 such that the solution of (2.13) satisfies
Ω
uε ≤ λ for all t ∈ (0, Tmax ),
t +τ
Ω
t
(3.1)
urε ≤ λ for all t ∈ (0, Tmax − τ )
(3.2)
urε ≤ λ,
(3.3)
and τ +1
0
Ω
where
1 τ := min 1, Tmax . 6
(3.4)
As a consequence of Lemma 3.1, let us establish the fundamental estimates associated with v . Firstly, due to (3.1) and Lemma 1.2 of [6], we have the following lemma: Lemma 3.2. There exists C > 0 such that the solution of (2.13) satisfies
Ω
|∇vε |l ≤ C for all t ∈ (0, Tmax )
with l ∈ [1, NN−1 ). Here we only state it as a lemma, while for the detailed proof readers can refer to [20] (see also [22]).
(3.5)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
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Lemma 3.3. Let p > 1 and β > 1. Assume that the initial data (u0 , v0 ) fulfills (2.5) and the diffusion function D satisfies (1.2)–(1.3). Then there exists a constant C > 0 such that
1 (β − 1) 2β ∇|∇vε |β 2 + 1 ∥uε ∥pLp (Ω ) + ∥∇vε ∥L2β (Ω ) + uεp+r −1 dt p 2β β2 2 Ω Ω (p − 1)CD 1 2β−2 2 2 2β |∇vε | | D vε | + |∇vε | + uεm+p−3 |∇ uε |2 +
d
1
2
≤C
Ω
4
Ω
χ (p − 1) 2
2CD
Ω
upε+1−m |∇vε |2 +
Ω
u2ε |∇vε |2β−2
Ω
+ C for all t ∈ (0, Tmax ).
(3.6)
Lemma 3.4. There exists C > 0 such that the solution of (2.13) satisfies
|∇vε |q ≤ C for all t ∈ (0, Tmax )
Ω
(3.7)
with q ∈ [1, (N +Nr ). 2−r )+ Proof. Firstly, let us pick the above τ ∈ (0, Tmax ). Then by the regularity principle asserted by Lemma 2.5, we have ¯ ) with ∂vε (·,τ ) = 0 on ∂ Ω , so that in particular, we can pick K > 0 such that (uε (·, τ ), vε (·, τ )) ∈ C 2 (Ω ∂ν
∥uε (s)∥L∞ (Ω ) ≤ K ,
∥vε (s)∥L∞ (Ω ) ≤ K and ∥∇vε (s)∥L∞ (Ω ) ≤ K for all s ∈ [0, τ ].
(3.8)
Next, integrating the second equation in (2.13) with respect to space, we have
d dt
Ω
vε (x, t ) +
vε (x, t ) =
Ω
Ω
uε (x, t ),
(3.9)
which together with (3.7) implies that
Ω
vε (x, t ) ≤ λ + ( v0 (x) − λ)e−t for all t ∈ (0, Tmax ),
(3.10)
Ω
and choose some α > where λ is the same as Lemma 3.1. Now, we fix q < (N +Nr 2−r )+ q<
N 2
α−
1 2
1 2
+
+
N 2r
−
r −1 r
1 2
such that
,
(3.11)
which implies that
−α −
N
2
1 r
−
1
q
r r −1
> −1.
(3.12)
Now, involving the variation-of-constants formula for vε , we have
vε (t + τ ) = e−τ (A+Id ) vε (t ) +
t +τ
e−(t +τ −s)(A+Id ) uε (s)ds,
t ∈ (0, Tmax − τ ).
(3.13)
t
Hence, it follows from (2.3), (3.10), (3.13) and the Young’s inequality that
∥(A + Id )α vε (t + τ )∥Lq (Ω ) t +τ −α− N2 1− 1q −α− N2 1r − 1q −µ(t +τ −s) ≤ C (q) e ∥uε (s)∥Lr (Ω ) ds + c τ ∥vε ∥L1 (Ω ) (t + τ − s) t
≤ C (q)
t +τ
t
+ cτ
−α− N2
≤ C (q)
−α− N2
1 1 r −q
r r −1
− rr−µ1 (t +τ −s)
(t + τ − s) e 1− 1q λ+ v0 (x) − λ e−t
r −r 1
1r
t +τ
∥uε (s)∥
ds t
r Lr (Ω ) ds
Ω
+∞
σ 0
−α− N2
1 1 r −q
r r −1
− rr−µ1 σ
e
dσ
r −r 1
1
λ r + cτ
−α− N2 1− 1q
λ+ v0 (x) − λ e−t . Ω
(3.14)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
Hence, due to (3.12), (3.14) and (2.2), we have
Ω
|∇vε (t + τ )|q ≤ C for all t ∈ (0, Tmax − τ )
(3.15)
for all q ∈ [1, (N +Nr ). Finally, in view of (3.8) and (3.15), we can get (3.7). 2−r )+ Lemma 3.5. Assume that m ≥ 1, κ ≥ max{ N2 ,
max m + 1, 2 −
2 N
−m
γ 2
}, β > 1 and γ < max{ NN−1 , (N +Nr }. If 2−r )+
N (2m − 2)κ − N + 2κ N 2 (2κ − γ )+
(2N β − N γ + 2γ ) + 2 −
2 N
− m,
then for all small δ > 0, we can find a constant C := C (p, β, γ , δ) > 0 such that
Ω
uεp+1−m |∇vε |2 ≤ δ
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C for all t ∈ (0, Tmax ). L (Ω )
(3.16)
Ω
Proof. By the Hölder inequality, we have
J1 :=
uεp+1−m |∇vε |2
Ω
′ uκε (p+1−m)
≤ Ω
κ1 Ω
p+m−1 2(p+1−m) 2 p+m−1 2κ ′ (p+1−m) = uε p+m−1 L
(Ω )
|∇vε |
2κ
1′ κ
∥∇vε ∥2L2κ (Ω ) ,
(3.17)
where κ and κ ′ satisfy κ1′ + κ1 = 1. Since, m ≥ 1, κ ≥ 2
κ
p+1−m
≤
p+m−1
p+m−1κ −1
≤
N N −2
N 2
and p ≥ 1 + m, we have
,
which together with Lemma 2.2 implies that
p+m−1 2(p+1−m) 2 p+m−1 uε 2κ ′ (p+1−m) p+m−1 L
(Ω )
p+m2 −1 µ1 ≤ C4 ∇ uε 2
L (Ω )
p+m−1 1−µ1 2 uε p+m2 −1 L
(Ω )
p+m−1 2 + uε
2(pp++m1−−1m) 2
L p+m−1 (Ω )
p+m−1 2(p+1−m)µ1 p+m−1 2 ≤ C5 + 1 ∇ uε
(3.18)
L2 (Ω )
with some positive constants C4 , C5 and
κ1 =
N [p+m−1] 2
1−
N 2
N (p+m−1) 2κ ′ (p+1−m)
− +
N [p+m−1] 2
= [p + m − 1]
N 2
−
1−
N 2
N 2κ ′ (p+1−m)
+
N [p+m−1] 2
∈ (0, 1). γ
On the other hand, due to Lemma 2.2, Lemma 3.2 and Lemma 3.4 and the fact that β ≤ 2βκ ≤
2N , N −2
we have
2
∥∇vε ∥2L2κ (Ω ) = ∥ |∇vε |β ∥ β2κ
L β (Ω )
2κ2 2(1−κ2 ) 2 ≤ C6 ∥∇|∇vε |β ∥L2β(Ω ) ∥ |∇vε |β ∥ γ β + ∥|∇vε |β ∥ βγ L β (Ω )
2κ2 β β ≤ C7 ∥∇|∇vε | ∥L2 (Ω ) + 1 , with some positive constants C6 , C7 and Nβ
κ2 =
γ
1−
− N 2
Nβ 2κ
+
Nβ
γ
N
=β
γ
1−
− N 2
N 2κ
+
Nβ
γ
∈ (0, 1).
L β (Ω )
(3.19)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2611
Inserting (3.18)–(3.19) into (3.17) and using the Young’s inequality, we have
p+m−1 2(p+1−m)κ1 2κ2 p+m−1 2 ∥∇|∇vε |β ∥ 2β + 1 J1 ≤ C8 ∇ u + 1 ε 2 L (Ω ) L (Ω )
1 1 N γ2 − κ p+m−1 N (p+1−m− κ ′ ) N Nβ 1− N + N [p+m−1] β 1− 2 + γ 2 2 2 + 1 = C8 + 1 ∇ uε 2 ∥∇|∇vε | ∥L2 (Ω ) L (Ω )
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C9 for all t ∈ (0, Tmax ). ≤δ L (Ω )
(3.20)
Ω
Here we have use the fact that p <
N (2m−2)κ−N +2κ N 2 (2κ−γ )+
(2N β − N γ + 2γ ) + 2 −
2 N
− m and p > 2 −
γ
2 N
− m.
Nβ
Lemma 3.6. Assume that β > 1, γ < max{ NN−1 , (N +Nr } and [γ −2(β−1)]+ ≥ θ ≥ 2β−2+N . If 2−r )+
p > max 1, 2 −
2
2θ (N − 1)
− m, 1 − m +
N
N
,
2θ − 1 2θ γ + 2N θ − N γ
(2N β − N γ + 2γ ) + 2 −
2 N
−m ,
then for all small δ > 0, we can find a constant C := C (p, θ , γ , β, δ) > 0 such that
Ω
u2ε |∇vε |2β−2 ≤ δ
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C for all t ∈ (0, Tmax ). L (Ω )
(3.21)
Ω
Proof. Firstly, due to the Hölder inequality, we have
J2 :=
Ω
u2ε |∇vε |2β−2
≤ Ω
u2ε θ
θ1 Ω
p+m−1 4 2 p+m−1 = uε p+4mθ−1 L
|∇vε |(2β−2)θ
′
1′ θ
(2β−2)
(Ω )
∥∇vε ∥ (2β−2)θ ′ L
(Ω )
,
(3.22)
where θ and θ ′ satisfy θ1 + θ1′ = 1. 2θ(N −1) On the other hand, with the help of p > 1 − m + and Lemma 2.2, we conclude that N
p+m−1 4 2 p+m−1 uε p+4mθ−1 L
(Ω )
p+m2 −1 κ3 ≤ C10 ∇ uε 2
p+m−1 (1−κ3 ) 2 uε p+m2 −1
L (Ω )
L
(Ω )
p+m−1 2 + uε
p+m4 −1 2
L p+m−1 (Ω )
p+m−1 4κ3 p+m−1 2 ≤ C11 + 1 2 ∇ uε
(3.23)
L (Ω )
with some positive constants C10 , C11 and
κ3 =
N [p+m−1] 2 N 1 2
−
N (p+m−1) 4θ N [p+m−1] 2
−
+
= [p + m − 1]
1−
N 2 N 2
N 4θ N [p+m−1] 2
− +
∈ (0, 1).
γ Nβ On the other hand, it then follows from 2(β−1) ≤ θ ′ ≤ (N −2)(β−1) , Lemma 2.2, Lemma 3.2 and Lemma 3.4 that (2β−2)
∥∇vε ∥ (2β−2)θ ′ L
(Ω )
= ∥ |∇vε |β ∥
2β−2
L
β (2β−2)θ ′ β (Ω )
(2β−2)κ4 (2β−2)(1−κ4 ) (2β−2) ≤ C12 ∥∇|∇vε |β ∥L2 (Ωβ) ∥ |∇vε |β ∥ γ β + ∥ |∇vε |β ∥ γ β L β (Ω )
(2β−2)κ4 β
≤ C13 ∥∇|∇vε |β ∥L2 (Ω )
L β (Ω )
+1
(3.24)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
with some positive constants C12 , C13 and Nβ
γ
κ4 =
Nβ (2β−2)θ ′ N + Nγβ 2
−
1−
N
=β
γ
−
1−
N
(2β−2)θ ′ N + Nγβ 2
∈ (0, 1).
θ−1 Inserting (3.23)–(3.24) into (3.22) and using p > 2θγ +22N (2N β − N γ + 2γ ) + 2 − θ−N γ
1
2β−2
1
2 N
− m and Lemma 2.2, we have
N γ − θ′ p+m−1 N 2− θ Nβ 1− N + N [p+m−1] 1− N + γ 2 β 2 2 2 + 1 J2 ≤ C14 + 1 2 ∇ uε ∥∇|∇vε | ∥L2 (Ω )
L (Ω )
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C15 for all t ∈ (0, Tmax ). ≤δ L (Ω )
(3.25)
Ω
Lemma 3.7. Assume that m satisfies (1.6). Then there exists number β¯ > 2, such that
∥uε (·, t )∥Lp (Ω ) + ∥∇vε (·, t )∥L2β (Ω ) ≤ C for all p ≥ 1 and β > β¯ . Proof. The proof of this lemma can be divided into three parts. Case r < NN+2 : by m > 2 − N2 , there exists 1 ≤ l0 < NN−1 such that
(N + l0 )(m − 1) − (N − l0 ) > 0. Nl (m−1)−(N −l )
Thus, choosing β¯ ≥ max{3N , (N +0l )(m−1)−(N −0 l ) }, γ = l0 , κ = 0 0
max 1, 2 −
2
− m, 1 − m +
N
2θ (N − 1)
N
N 2
and θ =
2
= max 1, 2 − ≤
N
β 2
, then
− m, 1 − m +
β(N − 1)
N
β −1 2 (2N β − Nl0 + 2l0 ) + 2 − − m β l0 + N β − Nl0 N
(3.26)
and N (2m − 2)κ − N + 2κ N2
(2κ − l0
)+
(2N β − Nl0 + 2l0 ) + 2 −
for all β ≥ β¯ . Also due to m > 2 −
2 N
2 N
−m=
m−1 N − l0
(2N β − Nl0 + 2l0 ) + 2 −
2 N
− m.
(3.27)
Nl (m−1)−(N −l )
and β¯ ≥ max{3N , (N +0l )(m−1)−(N −0 l ) }, we have 0 0
β −1 2 m−1 2 (2N β − Nl0 + 2l0 ) + 2 − − m < (2N β − Nl0 + 2l0 ) + 2 − − m β l0 + N β − Nl0 N N − l0 N
(3.28)
for all β ≥ β¯ . Now, choosing δ small enough in Lemmas 3.5–3.6, then the solution of (2.13) satisfies that d dt
Ω
upε +
Ω
1 |∇vε |2β + r upε + |∇vε |2β ≤ C26 , 2
Ω
(3.29)
Ω
which implies that d dt
y(t ) + C27 y(t ) ≤ C28 ,
where y := Ω upε + Ω |∇vε |2β . Thus a standard ODE comparison argument implies boundedness of y(t ) for all t ∈ (0, Tmax ). Clearly, ∥uε (·, t )∥Lp (Ω ) and ∥∇vε (·, t )∥L2β (Ω ) are bounded for all t ∈ (0, Tmax ). Finally, with the help of the Hölder inequality, we can get the results. 2 Case N + ≥ r ≥ NN+2 : by m > 2 − N2r+2 , there exists NN−1 ≤ q0 < N +Nr2−r such that 2
(N + q0 )(m − 1) − (N − q0 ) > 0. Nq (m−1)−(N −q )
Thus, choosing β¯ ≥ max{3N , (N +q0 )(m−1)−(N −0q ) }, γ = q0 , κ = 0 0
max 1, 2 −
2 N
− m, 1 − m +
2θ (N − 1) N
N 2
≥
= max 1, 2 −
q0 2
2 N
and θ =
β 2
, then
− m, 1 − m +
β(N − 1)
N
β −1 2 ≤ (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N
(3.30)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2613
and N (2m − 2)κ − N + 2κ N2
(2κ − q0
)+
(2N β − Nq0 + 2q0 ) + 2 −
2 N
−m=
m−1 N − q0
(2N β − Nq0 + 2q0 ) + 2 −
2 N
−m
(3.31)
Nq (m−1)−(N −q )
for all β ≥ β¯ . Also due to m > 2 − N2r and β¯ ≥ max{3N , (N +q0 )(m−1)−(N −0q ) }, we have +2 0 0
β −1 2 m−1 2 (2N β − Nq0 + 2q0 ) + 2 − − m < (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N N − q0 N for all β ≥ β¯ . Employing the same arguments as in the proof of case r < Case r > q0 ≥
N +2 : 2
N 2
There exists N −1 ≤ q0 < N
Nr (N +2−r )+
N +2 , N
(3.32)
we conclude the results.
such that
.
Thus, choosing β¯ ≥ 3N, γ = q0 , κ =
max 1, 2 −
2 N
− m, 1 − m +
and θ =
β
2θ (N − 1)
q0 2
2
N
, then obviously,
2 β(N − 1) = max 1, 2 − − m, 1 − m + N
N
β −1 2 ≤ (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N N (2m − 2)κ − N + 2κ 2 < (2N β − Nq0 + 2q0 ) + 2 − − m 2 + N (2κ − q0 ) N = +∞ for all β ≥ β¯ and m ≥ 1. Invoking the same arguments as in the proof of case r < of Lemma 3.7 is complete.
N +2 , N
(3.33)
we conclude the results. The proof
Underlying the estimates established above, we can derive the following boundedness results by invoking a Moser-type iteration (see Lemma A.1 in [6]) and standard parabolic regularity arguments. Lemma 3.8. Let (1.6) hold. Then one can find (εj )j∈N ⊂ (0, 1) such that εj → 0 as j → ∞ and that
∥uε (·, t )∥L∞ (Ω ) ≤ C for all t ∈ (0, Tmax )
(3.34)
∥vε (·, t )∥W 1,∞ (Ω ) ≤ C for all t ∈ (0, Tmax ).
(3.35)
and
By virtue of (2.15) and Lemma 3.8, the local-in-time solution can be extended to the global-in-time solution. Lemma 3.9. Assume that m satisfies (1.6). Then one can find C > 0 independent of ε ∈ (0, 1) such that
∥uε (·, t )∥L∞ (Ω ) ≤ C for all t ∈ (0, ∞)
(3.36)
∥vε (·, t )∥W 1,∞ (Ω ) ≤ C for all t ∈ (0, ∞).
(3.37)
and
As a straightforward result of Lemma 3.9, the following lemma gives uniform Hölder regularity properties of vε , ∇vε and uε . Here we only state the lemma, as for the detailed proof, readers can refer to the arguments of Lemma 3.18 and Lemma 3.19 in [22]. Lemma 3.10. Suppose that (1.6) is valid. Then one can find ζ ∈ (0, 1) such that for some C > 0
∥uε (·, t )∥
C
ζ ζ, 2
(Ω ×[t ,t +1])
≤ C for all t ∈ (0, ∞)
(3.38)
and such that for any τ > 0 there exists C (τ ) > 0 fulfilling
∥∇vε (·, t )∥
C
ζ ζ, 2
(Ω ×[t ,t +1])
≤ C for all t ∈ (τ , ∞).
(3.39)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
4. Regularity properties of time derivatives With the help of the estimates established before, we can obtain the boundedness property of the time derivatives of certain powers of uε on a fixed finite time interval, which will contribute to passing to the limit in the first equation in (2.13). Employing almost exactly the same arguments as in the proof of Lemmas 3.22–3.23 in [22] we can conclude the following lemmas. Since the procedure is rather standard, we may confine ourselves to an outline. Lemma 4.1. Assume that m satisfies (1.6) and p > 1 satisfies that p ≥ m − 1. Then for any fixed T > 0, there exists C (T ) > 0 with the property that T
p+m−1 2 ∇ uε 2 ≤ C (T )
(4.1)
Ω
0
for all ε ∈ (0, 1). Proof. Firstly, with the help of Lemma 3.10, for all ε ∈ (0, 1), we can fix a positive constant C1 such that and |∇vε | ≤ C1 in Ω × (0, ∞).
uε ≤ C 1
(4.2)
Let ς > m, ς ≥ 2(m − 1) and p := ς − m + 1. Then with the help of ς > m and ς ≥ 2(m − 1), we derive that p > 1 and p ≥ m − 1. Now, taking upε−1 as the test function for the first equation of (2.13) and using the Young’s inequality, we obtain 1 d p dt
∥uε ∥pLp (Ω ) + (p − 1)
Ω
upε−2 Dε (uε )|∇ uε |2 ≤ χ (p − 1)
Ω
≤ χ (p − 1) ≤
p−1
Ω
CD (p − 1)
uε
2
+µ
upε−1 ∇ uε · ∇vε + µ
Ω
p
uε −
Ω
|∇ uε | |∇vε | + µ
uεm+p−3 |∇ uε |2 +
p+r −1
uε
Ω
Ω
upε −
p
Ω
uεp+r −1
Ω
uε −
χ 2 (p − 1)
2CD
uεp+r −1
Ω
Ω
upε+1−m |∇vε |2
,
(4.3)
which implies that 1 d p dt
p uε Lp (Ω )
∥ ∥
+
CD (p − 1)
2
Ω
+p−3 um |∇ uε |2 ≤ ε
χ 2 (p−1)
Ω
2CD
uεp+1−m |∇vε |2 + µ
Ω
upε −
Ω
uεp+r −1 .
(4.4)
Now, in view of (4.2), we integrate (4.4) with respect to t over (0, T ) for some fixed T > 0 and then have 1
p
Ω
upε (·, T ) +
CD (p − 1)
T
2
0
Ω
+p−3 um |∇ uε |2 ≤ ε
≤
χ 2 (p − 1) 2CD
χ 2 (p − 1) 2CD
T
0
p
Ω
p+3−m
C1
uεp+1−m |∇vε |2 + µC1 T |Ω | + p
T |Ω | + µC1 T |Ω | +
1 p
p
Ω
u0 .
1 p
p
Ω
u0 (4.5)
On the other hand, by p = ς − m + 1, we have T
Ω
0
2 uς− |∇ uε |2 = ε
T
0
Ω
+p−3 um |∇ uε |2 ≤ C2 (1 + T ) ε
for some positive constant C2 . The proof of Lemma 4.1 is completed.
(4.6)
Lemma 4.2. Let (1.6) hold. Then one can find ε ∈ (0, 1) such that for some C > 0
∥∂t uε (·, t )∥(W 2,2 (Ω ))∗ ≤ C for all t ∈ (0, ∞).
(4.7)
0
Moreover, let ς > m and ς ≥ 2(m − 1). Then for all T > 0 and ε ∈ (0, 1) there exists C (T ) > 0 such that T
0
∥∂t uςε (·, t )∥(W lN ,2 (Ω ))∗ dt ≤ C (T ) for all t ∈ (0, ∞) and lN > 0
N +2 2
and l ∈ N.
(4.8)
Proof. Firstly, with the help of (4.2) and Dε ≤ D + 2ε (for all ε ∈ (0, 1)), we have Dε (uε ) ≤ C3
in Ω × (0, ∞) for all ε ∈ (0, 1),
where C3 := ∥D∥L∞ ((0,C1 )) + 2.
(4.9)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2615
Now, testing the first equation by certain ϕ ∈ C0∞ (Ω ), we have
Ω
uε,t (·, t )ϕ =
Ω
= Ω
where Hε (s) :=
s
[∇ · (Dε (uε )∇ uε ) − χ∇ · (uε ∇vε )] ϕ +
Hε (uε )1ϕ + χ
uε ∇vε · ∇ϕ + µ
Ω
Ω
Ω
µuε − urε ϕ
uε ϕ −
Ω
urε ϕ
for all t ∈ (0, ∞),
(4.10)
Dε (τ )dτ for s ≥ 0. On the other hand, by Dε ≤ D + 2ε and uε ≥ 0, we also derive
0
Hε (uε ) ≤ C3 := C1 (∥D∥L∞ ((0,C1 )) + 2)
in Ω × (0, ∞) for all ε ∈ (0, 1).
(4.11)
Therefore, by (4.2) and (4.9)–(4.11), we obtain
r −1 2 uε,t (·, t )ϕ ≤ C3 |1ϕ| + C1 χ |∇ϕ| + C1 (µ + C1 ) |ϕ| Ω
Ω
Ω
(4.12)
Ω
for all t ∈ (0, ∞) and for all ε ∈ (0, 1). Thus, (4.12) yields (4.7). 1 Now, for any fixed ψ ∈ C0∞ (Ω ), multiplying the first equation by uς− ψ , we have ε 1
ς
Ω
∂t uςε (·, t )ψ =
Ω
uςε −1 [∇ · (Dε (uε )∇ uε ) − χ∇ · (uε ∇vε )] ψ +
= −(ς − 1)
Ω
+ χ (ς − 1) +µ
Ω
uςε −2 Dε (uε )|∇ uε |2 ψ
uςε ψ
− Ω
Ω
− Ω
r −1 uς+ ψ ε
1 uς− µuε − urε ψ ε
Ω
uςε −1 Dε (uε )∇ uε
uςε −1 ∇ uε · ∇vε ψ + χ
Ω
· ∇ψ
uςε ∇vε · ∇ψ
for all t ∈ (0, ∞).
(4.13)
Next, we will estimate the right-hand sides of (4.13). To this end, assuming that p := ς − m + 1, then ς > m and ς ≥ 2(m − 1) yield to p > 1 and p ≥ m − 1. Now, with the help of (4.2), we also derive that
µ
Ω
uςε ψ −
ς
Ω
uςε +r −1 ψ ≤ C1 |Ω |(µ + C1r −1 )∥ψ∥L∞ (Ω )
for all ε ∈ (0, 1).
(4.14)
Moreover, by (4.1), (4.13)–(4.14) and the Young’s inequality and applying the same arguments as in the proof of Lemma 3.22 in [22] we conclude that there exists C4 > 0 such that
ς−2 2 ∂t uς (·, t )ψ ≤ C4 u |∇ u | + 1 ∥ψ∥W 1,∞ (Ω ) . ε ε ε Ω
(4.15)
Ω
We choose lN ∈ N large enough to satisfy lN > exists C5 > 0 such that
∥∂t uςε (·, t )∥(W lN ,2 (Ω ))∗ ≤ C5
0
N +2 2
l ,2
and hence W0N (Ω ) ↩→ W 1,∞ (Ω ). Hence, (4.15) implies that there
Ω
uςε −2 |∇ uε |2 + 1
Now, combining (4.6) and (4.16), we can get (4.8).
for all t ∈ (0, ∞) and any ε ∈ (0, 1).
(4.16)
Lemma 4.3. Assume that m satisfies (1.6) Then there exists (εj )j∈N ⊂ (0, 1) such that εj → 0 as j → ∞ and that uε → u
a.e. in Ω × (0, ∞),
uε ⇀ u
weakly star in L (Ω × (0, ∞)), 2 ,2 W0
in
0 Cloc
([0, ∞); (
vε → v in
0 Cloc
¯ × [0, ∞)), (Ω
uε → u
(4.17)
∞
(Ω )) ), ∗
(4.18) (4.19) (4.20)
¯ × [0, ∞)) (Ω
(4.21)
∇vε → ∇v in L∞ (Ω × (0, ∞))
(4.22)
∇vε → ∇v in
0 Cloc
and
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
with some triple (u, v) which is a global weak solution of (2.13) in the sense of Definition 2.1. Moreover, u satisfies 0 u ∈ Cω−∗ ([0, ∞); L∞ (Ω )).
(4.23)
Proof. Firstly, by the same arguments as in the proof of Lemma 4.1 in [22] we can get that (4.19)–(4.23). On the other hand, by Lemma 3.9, we derive that for certain u ∈ L∞ (Ω × (0, ∞)), (4.18) holds. Moreover, we next fix ς > m satisfying ς ≥ 2(m − 1) and set p := 2ζ − m + 1, then by (4.6), we have for each T > 0, (uςε )ε∈(0,1) is bounded in L2 ((0, T ); W 1,2 (Ω )). Furthermore, Lemma 4.2 implies that
(∂t uςε )ε∈(0,1)
l ,2
is bounded in L1 ((0, T ); (W0N (Ω ))∗ ) for each T > 0 and lN >
N +2 2
.
Hence, by Aubin–Lions lemma (Theorem III.2.3 of [23]), we also derive that the strong precompactness of (uςε )ε∈(0,1) in L2 (Ω × (0, T )). Thus, we can pick a suitable subsequence such that uςε → z ς for some nonnegative measurable z : Ω × (0, Ω ) → R. In view of (4.18) and the Egorov theorem, we have z = u necessarily, so that (4.17) is valid. Now, we will prove that (u, v) is a global weak solution of (2.13) in the sense of Definition 2.1. To this end, by (4.17), (4.18), (4.20)–(4.22) we have (2.9)–(2.10) hold and u and v are nonnegative functions. Finally, letting ε := εj ↘ 0 and using (4.17)–(4.22), we can get (2.11)–(2.12). The proof of Lemma 4.3 is completed. Now we can easily prove Theorem 2.1. Proof of Theorem 2.1. Combining Lemma 3.9 with Lemma 4.3, we prove Theorem 2.1. 5. Asymptotic behavior In this section we study the long-time behavior for (2.13) in the case µ = 0. As the first step, we give the decay property separately for the integrals of the solution components u and v . Lemma 5.1. Let µ = 0 and m satisfy (1.6). Then we have
u( x , t ) ≤
Ω
u0 (x)
Ω
1−r
1−1 r + (r − 1)|Ω |
1−r
for all t ∈ (0, ∞).
t
(5.1)
Proof. Let t > 0 and s ∈ (0, t ). Since µ = 0, it follows from an integration by parts to the first equation in (2.13) and the Hölder inequality that d
ds
Ω
u(x, s) = −
u (x, s) ≤ −|Ω |
1 −r
r
Ω
Ω
u(x, s)
r
for all s ∈ (0, t ),
(5.2)
which implies that
Ω
1−r 1−r u(x, t ) − u0 (x) ≥ (r − 1)|Ω |1−r t .
Hence, (5.1) holds.
(5.3)
Ω
As a consequence, we obtain a basic decay property also for the second solution component. Lemma 5.2. There exists C > 0 such that
C
v(x, t ) ≤
for all t ∈ (0, Tmax ). (5.4) 1 (1 + t ) r −1 Proof. Firstly, let z (t ) := Ω v(x, t ), t ∈ [0, Tmax ). Next, integrating the second equation in (2.13) we see that there exists C1 > 0 such that ′ z (t ) = −z (t ) + u(x, t ), Ω
Ω
1
≤ −z (t ) + C1 (1 + t ) 1−r for all t ∈ (0, Tmax ). 1 min{1, r − } 1 max{ We now let C2 := 2 v (x), 2} and define Ω 0 1 − min{1, r − } 1
z¯ (t ) := C2 (t + 2)
for all t ≥ 0.
(5.5)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
Then z¯ (0) =
C2 min{1, 1 } r −1
≥
2
Ω
2617
v0 (x) = z0 and 1
1
− min 1,
1
−1
− min 1,
1
1
r −1 r −1 + C2 (t + 2) − C1 (1 + t ) 1−r (t + 2) r −1 1 1 1 −1 1 − min 1, r − 1 ≥ C2 (t + 2) − min 1, 2 t +2 r −1 1 1 1 1 − min 1, r − min 1 , r− 1 1 1 − r C2 − 2C1 (t + 2) (1 + t ) + (t + 2) 2 1 1 − min 1, r − −1 1 1 ≥ C2 (t + 2) − 2 2 1 1 1 1 1 − min 1, r − 1+min 1, r − min 1, r − + 1− 1 1 1 r + (t + 2) C2 − C1 2 (t + 1)
z¯ ′ (t ) + z¯ (t ) − C1 (1 + t ) 1−r = −C2 min 1,
2
≥ 0 for all t > 0.
(5.6)
By comparison, we thus infer that z (t ) ≤ z¯ (t ) for all t ∈ (0, Tmax ), which directly establishes (5.4).
Now thanks to the precompactness of both (u(·, t ))t >1 and (v(·, t ))t >1 implied by Lemmas 4.1 and 4.2, the latter two decay properties readily entail the first of our main results on large time behavior, addressing the case when µ = 0. Lemma 5.3. Let µ = 0 and m satisfy (1.6). Then with (u, v) as given by Theorem 2.1, we have ∗
u(·, t ) ⇀ 0 in L∞ (Ω ) as t → ∞.
(5.7)
Proof. We suppose that (5.7) is not valid. Then there exists a sequence (tk )k∈N such that tk → ∞ as k → ∞, and that for ˜ ∈ L1 ( Ω ) certain ψ
Ω
˜ ≥ C1 u(x, tk )ψ
for all k ∈ N
(5.8)
with some C1 > 0. From Lemma 3.9, we can find C2 > 0 such that u(x, t ) ≤ C2
for a.e. (x, t ) ∈ Ω × (0, ∞).
(5.9)
˜ L1 (Ω ) ≤ Due to the density of C0∞ (Ω ) in L1 (Ω ), we can pick a ψ ∈ C0∞ (Ω ) such that ∥ψ − ψ∥ that Ω
u(x, tk )ψ ≥
≥
Ω
C1 , 2C2
thus it follows from (5.9)
˜ − ∥u(·, tk )∥L∞ (Ω ) ∥ψ − ψ∥ ˜ L1 (Ω ) u(x, tk )ψ
C1
for all k ∈ N .
2
(5.10)
Now, in light of (4.7), there exists C3 > 0 fulfilling
∥uε (·, t ) − uε (·, s)∥(W 2,2 (Ω ))∗ ≤ C3 |t − s| for all t , s ∈ (0, ∞) and ε ∈ (0, 1),
(5.11)
0
which along with convergence (4.19) yields
∥u(·, t ) − u(·, s)∥(W 2,2 (Ω ))∗ ≤ C3 |t − s| for all t , s ∈ (0, ∞).
(5.12)
0
If we let τ ∈ (0, 1) such that
τ≤
C1 4C3 ∥ψ∥W 2,2 (Ω )
,
0
then for all t ∈ (tk , tk + τ ) with k ∈ N we have
u(x, tk )ψ − ≤ ∥u(·, t ) − u(·, t )∥ 2,2 ∗ ∥ψ∥ 2,2 u ( x , t )ψ (W0 (Ω )) W0 (Ω ) Ω
Ω
≤ C3 |tk − t |∥ψ∥W 2,2 (Ω ) 0
≤
C1 4
for all t ∈ (tk , tk + τ ) with k ∈ N
(5.13)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
which combined with (5.10) entails
Ω
C1
u(x, t )ψ ≥
4
for all t ∈ (tk , tk + τ ) with k ∈ N .
(5.14)
We set uk (x, s) := u(x, tk + s), (x, s) ∈ Ω × (0, τ ) for all k ∈ N . Then an application of (5.14) to uk yields
Ω
C1
uk (x, s)ψ ≥
4
for all k ∈ N and s ∈ (0, τ ).
(5.15)
Additionally, from Lemma 5.1, we can find C4 > 0 satisfying
Ω
uk ( x , s ) =
Ω
u( x , t k + s ) ≤
C4 1
(1 + tk + s) r −1
for all k ∈ N .
(5.16)
Case r = 2: we integrate (5.15) with respect to s over (0, τ ) and then obtain C1 τ 4
τ
≤ 0 τ Ω ≤ Ω
0
≤ C4 ln
uk (x, s)ψ(x)dxds C4 ψ(x) 1 + tk + s
1 + tk + τ 1 + tk
dxds
Ω
ψ(x)dx → 0 as k → ∞,
(5.17)
which is absurd and thereby proves that virtually (5.7) is valid. Case r ̸= 2: integrate (5.15) with respect to s over (0, τ ) and then obtain C1 τ 4
τ
≤ 0 τ Ω ≤
uk (x, s)ψ(x)dxds C4 ψ(x)
dxds
1
(1 + tk + s) r −1 2−r 2−r 1−r ψ(x)dx → 0 as k → ∞, ≤ C4 (1 + tk + τ ) 1−r − (1 + tk ) 1−r 2−r Ω 0
Ω
which is absurd and thereby proves that virtually (5.7) is valid.
(5.18)
With the help of Lemmas 3.10 and 5.2, we can achieve the decay property of v : Lemma 5.4. Let µ = 0 and m satisfy (1.6). Then with (u, v) as given by Theorem 2.1, we have
v(·, t ) → 0 in L∞ (Ω ) as t → ∞.
(5.19)
Proof. Assume that (5.19) does not hold, then we can find (tk )k∈N and C1 > 0 such that tk → ∞ as k → ∞ and
∥v(·, tk )∥L∞ (Ω ) ≥ C1 for all k ∈ N .
(5.20)
On the basis of Lemma 3.10, an application of the Arzela–Ascoli theorem allows for an extraction of a subsequence from v(·, tk )k∈N such that
v(·, tk ) → v∞ in L∞ (Ω ) as k → ∞
(5.21)
¯ ), where we still denote the subsequence by v(·, tk )k∈N . However, Lemma 5.2 shows with a nonnegative function v∞ ∈ C (Ω that v(·, t ) → 0 in L1 (Ω ) as t → ∞, thereby necessarily v∞ ≡ 0, which is contradiction to (5.20) and thus (5.19) is valid. 0
Proof of Theorem 2.2. The conclusions of Theorem 2.2 follow from Lemmas 5.3 and 5.4. Acknowledgments The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work. The first author is partially supported by the National Natural Science Foundation of China (No. 11601215) and partially supported by the Doctor Start-up Funding of Ludong University (No. LA2016006). The second author is partially supported by NNSFC (No. 11571363).
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source Jiashan Zheng a,∗ , Yifu Wang b a
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, PR China
b
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China
article
info
Article history: Received 11 March 2016 Received in revised form 5 August 2016 Accepted 17 September 2016 Available online 17 October 2016 Keywords: Boundedness Chemotaxis Global existence Decay behavior Logistic source
abstract This paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype
ut = ∆um − ∇ · (u∇v) + µu − ur , x ∈ , t > 0, vt = ∆v − v + u, x ∈ , t > 0, ∂v ∂u = = 0, x ∈ ∂ , t > 0, ∂ν ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x) x ∈ ,
(0.1)
in a smooth bounded domain ⊂ RN (N ≥ 2) with parameters m, r ≥ 1 and µ ≥ 0. The PDE system in (0.1) is used in mathematical biology to model the mechanism of chemotaxis, that is, the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space. It is shown that if
N +2 2 if 1 < r < , >2− N N + N +2 N +2 (N + 2 − 2r ) m >1+ if ≥r≥ , N +2 2 N ≥ 1 if r > N + 2 , 2
¯ ) × W 1,∞ ()(ι > 0), then (0.1) possesses and the nonnegative initial data (u0 , v0 ) ∈ C ι ( at least one global bounded weak solution. Apart from this, it is proved that if µ = 0 then both u(·, t ) and v(·, t ) decay to zero with respect to the norm in L∞ () as t → ∞. © 2016 Elsevier Ltd. All rights reserved.
∗
Corresponding author. E-mail address: [email protected] (J. Zheng).
http://dx.doi.org/10.1016/j.camwa.2016.09.020 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2605
1. Introduction In this paper, we consider the Neumann initial–boundary value problem for a full chemotaxis system with generalized volume-filling effect and nonlinear logistic source
ut = ∇ · (D(u)∇ u) − χ∇ · (S (u)∇v) + µu − ur , vt = 1v + u − v, x ∈ Ω , t > 0, ∂u ∂v = = 0, x ∈ ∂ Ω , t > 0, ∂ν ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ Ω
x ∈ Ω , t > 0, (1.1)
in a bounded domain Ω ⊂ RN (N ≥ 2) with smooth outward normal vector field ν on ∂ Ω , where S (u) = u is the sensitivity function, parameter χ > 0, and the initial data u0 ∈ C ι (Ω )(ι > 0) and v0 ∈ W 1,∞ (Ω ) are assumed to satisfy u0 , v0 ≥ 0 in Ω. The problem of this type is used to describe theoretically the process of chemotaxis, the biological phenomenon of the oriented movement of cells or organisms in response to a chemical signal. The pioneering works of chemotaxis model were introduced by Patlak [1] in 1953 and Keller and Segel [2] in 1970, and we refer the reader to the survey by Hillen and Painter [3] and Horstmann [4], where a comprehensive information of further examples illustrating the outstanding biological relevance of chemotaxis can be found. In this paper, we assume that the diffusion function D(u) satisfies θ D ∈ Cloc ([0, ∞)) for some θ > 0,
(1.2)
and there exist some constants m ≥ 1 and CD > 0 such that D(u) ≥ CD um−1
for all u ≥ 0.
(1.3)
During the past decades, the variants of (1.1) without cell kinetics have been studied extensively by researchers and the main issue of the investigation was whether the solutions of the models are bounded or blow-up (see e.g. Horstmann and Winkler [5], Tao and Winkler [6], Ishida et al. [7], Winkler [8]). In many applications, blow-up phenomena do not appropriately reflect the respective experimentally observable behavior. Accordingly, considerable efforts have been devoted to developing models in which blow-up phenomena are prevented. In this context it is recognized that cell proliferation terms of logistic type, such as contained in (1.1), form the possibly simplest among the blow-up preventing mechanisms. In fact, the presence of such logistic terms is sufficient to suppress any blow-up in many relevant situations. For example, if r = 2 (quadratic type), Winkler [9] discussed the global boundedness of classical solutions to problem (1.1) on a smooth bounded convex domain under the assumption that either N ≤ 2, or the logistic damping effect is large enough. A large number of facts indicate that Keller–Segel-growth systems with logistic-type kinetics of superlinear, not necessarily quadratic type. In fact, when D(u) = (u + 1)−α , the sensitivity function S (u) = u is replaced by S (u) = (u + 1)β−1 with 0 < α + β < N2 and the logistic term f satisfies f (u) ≤ µ − bur
for all u ≥ 0
(1.4)
with some µ ≥ 0, b > 0 and r = 2, Wang et al. [10] obtained the unique global uniformly bounded classical solution (u, v) of problem (1.1), but there is not any available result on the boundedness of the solution when α + β > N2 . Furthermore, assuming that the logistic source f ∈ C ∞ ([0, ∞)) satisfies f (u) ≤ µu − bu2
for all u ≥ 0
(1.5)
and the diffusion function D and the sensitivity function S fulfill D, S ∈ C 2 ([0, ∞)) and
D(s) ≥ 0 for all s ≥ 0,
c1 s ≤ D(s) for all s ≥ s0 , p
c1 sβ ≤ S (s) ≤ c2 sβ
for all s ≥ s0
with c2 > c1 > 0, s0 > 1 and p, β ∈ R, Cao [11] proved that if β < 1, then the classical solution of (1.1) is globally bounded, whereas the question whether or not blow-up may occur is left therein when β > 1. Recently, in [12], it is proved that if 0 < α + β < max{r − 1 + α, N2 } or b is big enough when β = r − 1, then the classical solutions to the corresponding system are uniformly bounded. There is also other alternative as the degenerate sensitivity function, degenerate problems have been studied to prevent blow up (see Chamoun et al. [13], Lorz [14]). Going beyond these boundedness statements, a number of results are available which show that the interplay of chemotactic cross-diffusion and cell kinetics of logistic-type may lead to quite a colorful dynamics. For instance, if D(u) ≡ 1, S (u) = u, f (u) = u − bu2 and the ratio χb is sufficiently
large, Winkler [15] proved that the unique nontrivial spatially homogeneous equilibrium given by u = v ≡ 1b is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data (u0 , v0 ) such that u0 ̸≡ 0, the problem (1.1) possesses a uniquely determined global classical solution (u, v) with (u, v)|t =0 = (u0 , v0 ) which satisfies
u(·, t ) −
b ∞
1
L
→ 0 and (Ω )
v(·, t ) −
b ∞
1
L
→0 (Ω )
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
as t → ∞. When N = 1, D(u) ≡ 1, S (u) = u, f (u) = µu − bu2 with b ≥ 1, Winkler [16] obtained that the solutions of parabolic–elliptic system (the second equation in (1.1) is replaced by an elliptic equation, i.e., 0 = 1v − v + u) may become large at intermediate time scales, thus exceeding the system’s carrying capacity to an arbitrary extent (though not blowing up). If f (u) = µu − buλ (λ > 1), µ, b ≥ 0 and D(u) ≡ 1, S (u) = u, N ≥ 5 and λ < 32 + 2N1−2 , Winkler [17] showed that there exist initial data such that the smooth local-in-time solution of parabolic–elliptic system blows up in finite time. In this paper accounting for the damping effect of logistic term more efficiently, we prove that if
N +2 2 if 1 < r < , >2− N N (N + 2 − 2r )+ N +2 N +2 m >1+ if ≥r≥ , N +2 2 N ≥ 1 if r > N + 2 ,
(1.6)
2 then for proper regularity hypotheses on the initial data, (1.1) possesses at least one globally bounded weak solution. Moreover, if µ = 0, then both u(·, t ) and v(·, t ) decay to zero with respect to the norm in L∞ (Ω ) as t → ∞. Our result quantitatively shows that when the effects of nonlinear diffusivity and logistic kinetics of cells are taken into account simultaneously, the range of relevant parameters can be enlarged, which can ensure the boundedness of the cell density. 4 Indeed, in the case β = 1, r = 2, our result implies that m > 2 − N + can ensure the existence of the bounded solutions, 2
whereas it is m > 2 − N2 in [10,12]. It is worth to remark the main idea underlying the proof of our results. Involving the variation-of-constants formula for v , we have
v(t + τ ) = e−τ (A+Id ) v(t ) +
t +τ
e−(t +τ −s))(A+Id ) u(s)ds,
t ∈ (0, Tmax − τ )
t
for some τ > 0 and the operator A defined below. Then with the help of boundedness of
Ω
T 0
Ω
ur dxdt, we can obtain
|∇v|σ ≤ C for all t ∈ (0, Tmax )
(1.7)
for σ ∈ [1, max{ (N +Nr , N }), and thereby establish the a priori estimates of the functional 2−r )+ N −1
p
2β
u +
|∇v|
Ω
Ω
for any p > 1 and β > 1.
2. Preliminaries and main results Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. To begin with, let us collect some basic solution properties which essentially have already been used in [5]. Lemma 2.1 ([5]). For p ∈ (1, ∞), let A := Ap denote the sectorial operator defined by Ap u := −1u
for all u ∈ D(Ap ) :=
∂ϕ ϕ ∈ W 2,p (Ω ) = 0 . ∂ν ∂ Ω
(2.1)
The fact that the spectrum of A is a p-independent countable set of positive real numbers 0 = µ0 < µ1 < µ2 < · · · entails the following consequences: (i) The operator A + Id possesses fractional powers (A + Id )α (α ≥ 0), the domains of which have the embedding properties D((A + Id )α ) ↩→ W 1,p (Ω )
if α >
1 2
.
(2.2)
(ii) Moreover, for all 1 ≤ p < q < ∞ and u ∈ Lp (Ω ) the general Lp − Lq estimate α −tA
∥(A + Id ) e
u∥Lq (Ω ) ≤ ct
−α− N2
1 1 p−q
e(1−µ)t ∥u∥Lp (Ω ) ,
for any t > 0 and α ≥ 0 with some µ > 0. Lemma 2.2 ([18]). Let 0 < θ ≤ p ≤
2N . N −2
There exists a positive constant CGN such that for all u ∈ W 1,2 (Ω ) ∩ Lθ (Ω ),
∥u∥Lp (Ω ) ≤ CGN (∥∇ u∥aL2 (Ω ) ∥u∥1Lθ−(aΩ ) + ∥u∥Lθ (Ω ) ) N
is valid with a =
N
θ−p 1− N + Nθ 2
∈ (0, 1).
(2.3)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2607
Lemma 2.3 ([19]). Let s ≥ 1 and q ≥ 1. Assume that p > 0 and a ∈ (0, 1) satisfy 1
−
2
p N
q
= (1 − a) + a s
1 2
−
1
and p ≤ a.
N
s
Then there exist c0 , c0′ > 0 such that for all u ∈ W 1,2 (Ω ) ∩ L q (Ω ),
∥u∥W p,2 (Ω ) ≤ c0 ∥∇ u∥aL2 (Ω ) ∥u∥1−s a
+ c0′ ∥u∥
L q (Ω )
s
L q (Ω )
.
Lemma 2.4 ([20]). Let T > 0, τ ∈ (0, T ), A > 0 and B > 0, and suppose that y : [0, T ) → [0, ∞) is absolutely continuous such that y′ (t ) + Ay(t ) ≤ h(t ) for a.e. t ∈ (0, T ) with some nonnegative function h ∈
L1loc
(2.4)
([0, T )) satisfying
t +τ
h(s)ds ≤ B
for all t ∈ (0, T − τ ).
t
Then
y(t ) ≤ max y0 + B,
B Aτ
+ 2B
for all t ∈ (0, T ).
Theorem 2.1. Let µ ≥ 0. Assume that the initial data (u0 , v0 ) are such that
¯ ) for certain ι > 0 with u0 ≥ 0 in Ω , u0 ∈ C ι ( Ω ¯, v0 ∈ W 1,∞ (Ω ) with v0 ≥ 0 in Ω
(2.5)
D and m satisfy (1.2) – (1.3) and (1.6), respectively. Then the problem (1.1) possesses at least one global weak solution (u, v) in the sense of Definition 2.1. This solution is bounded in Ω × (0, ∞) in the sense that
∥u(·, t )∥L∞ (Ω ) + ∥v(·, t )∥W 1,∞ (Ω ) ≤ C for all t > 0.
(2.6)
¯ × [0, ∞) and Furthermore, v is continuous in Ω 0 u ∈ Cω−∗ ([0, ∞); L∞ (Ω )),
(2.7)
that is, u is continuous on [0, ∞) as an L (Ω )-valued function with respect to the weak-∗ topology. ∞
Theorem 2.2. Let µ = 0, and suppose that D fulfills (1.2)–(1.3) with m satisfying (1.6). Then as long as (u0 , v0 ) fulfills (2.5) with u0 ̸≡ 0, the global weak solution (u, v) constructed in Theorem 2.1 satisfies ∗
∥u(·, t )∥L∞ (Ω ) ⇀ 0,
∥v(·, t )∥L∞ (Ω ) → 0
(2.8)
as t → ∞.
s
Definition 2.1. Let T > 0, (u0 , v0 ) fulfill (2.5) and H (s) = 0 D(σ )dσ for s ≥ 0. Then a pair (u, v) of nonnegative functions is called a weak solution of (1.1) if the following conditions are satisfied
¯ × [0, T )), u ∈ L1loc (Ω
(2.9)
1 1 ,1 ¯ v ∈ L∞ (Ω )), loc (Ω × [0, T )) ∩ Lloc ([0, T ); W
H (u)
¯ × [0, T )), and u|∇v| belong to L1loc (Ω
(2.10)
and T
− Ω
0
uϕt −
Ω
u0 ϕ(·, 0) =
T
H (u)1ϕ + χ
Ω
0
T
Ω
0
u∇v · ∇ϕ + µ
0
T
Ω
ϕu −
0
T
Ω
ϕ ur
(2.11)
∂ϕ
¯ × [0, T )) satisfying = 0 on ∂ Ω × (0, T ) as well as for any ϕ ∈ C0∞ (Ω ∂ν T
− 0
Ω
vϕt −
Ω
v0 ϕ(·, 0) = −
T
T
∇v · ∇ϕ − 0
Ω
0
Ω
vϕ +
T
0
Ω
uϕ
(2.12)
¯ × [0, T )). for any ϕ ∈ C0∞ (Ω If (u, v) is a weak solution of (1.1) in Ω × (0, T ) for all T > 0, then (u, v) is said to be a global weak solution of (1.1).
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
We intend to construct a weak solution of (1.1) as the limit of a sequence of solutions to the regularized systems. Thus, in this section, we firstly establish the regularized system of (1.1). To this end, we approximate the diffusion coefficient function in (1.1) by a family (Dε )ε∈(0,1) of functions Dε ∈ C 2 ((0, ∞))
such that Dε (n) ≥ ε for all n > 0
and D(n) ≤ Dε (n) ≤ D(n) + 2ε
for all n > 0 and ε ∈ (0, 1).
Therefore, the regularized problems of (1.1) are presented as follows
u = ∇ · (D (u )∇ u ) − χ∇ · (u ∇v ) + µu − ur , εt ε ε ε ε ε ε ε vεt = 1vε − vε + uε , x ∈ Ω , t > 0, ∂vε ∂ uε = = 0, x ∈ ∂ Ω , t > 0, ∂ν ∂ν uε (x, 0) = u0 (x), vε (x, 0) = v0 (x), x ∈ Ω .
x ∈ Ω , t > 0, (2.13)
The following local existence result on (2.13) is rather standard, and we like to refer the readers to [21,10,22] for its proof. Lemma 2.5 ([21,10,22]). For any ε ∈ (0, 1), there exist Tmax ∈ (0, ∞] and a classical solution (vε , vε ) of (2.13) in Ω ×(0, Tmax ) such that
¯ × [0, Tmax )) ∩ C 2,1 (Ω ¯ × (0, Tmax )), uε ∈ C 0 ( Ω 0 ¯ ¯ × (0, Tmax )). vε ∈ C (Ω × [0, Tmax )) ∩ C 2,1 (Ω
(2.14)
Moreover, uε and vε are nonnegative in Ω × (0, Tmax ), and
∥uε (·, t )∥L∞ (Ω ) + ∥vε (·, t )∥W 1,∞ (Ω ) → ∞ as t ↗ Tmax .
(2.15)
3. A priori estimates In this section, we are going to establish an iteration step to develop the main ingredient of our result. The iteration depends on a series of a priori estimates. Now, due to the presence of the nonlinear logistic source in the first equation of (1.1), some useful estimates can be derived from Lemma 2.2 in [20], which can be stated in the following. Lemma 3.1. There exists λ > 0 such that the solution of (2.13) satisfies
Ω
uε ≤ λ for all t ∈ (0, Tmax ),
t +τ
Ω
t
(3.1)
urε ≤ λ for all t ∈ (0, Tmax − τ )
(3.2)
urε ≤ λ,
(3.3)
and τ +1
0
Ω
where
1 τ := min 1, Tmax . 6
(3.4)
As a consequence of Lemma 3.1, let us establish the fundamental estimates associated with v . Firstly, due to (3.1) and Lemma 1.2 of [6], we have the following lemma: Lemma 3.2. There exists C > 0 such that the solution of (2.13) satisfies
Ω
|∇vε |l ≤ C for all t ∈ (0, Tmax )
with l ∈ [1, NN−1 ). Here we only state it as a lemma, while for the detailed proof readers can refer to [20] (see also [22]).
(3.5)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2609
Lemma 3.3. Let p > 1 and β > 1. Assume that the initial data (u0 , v0 ) fulfills (2.5) and the diffusion function D satisfies (1.2)–(1.3). Then there exists a constant C > 0 such that
1 (β − 1) 2β ∇|∇vε |β 2 + 1 ∥uε ∥pLp (Ω ) + ∥∇vε ∥L2β (Ω ) + uεp+r −1 dt p 2β β2 2 Ω Ω (p − 1)CD 1 2β−2 2 2 2β |∇vε | | D vε | + |∇vε | + uεm+p−3 |∇ uε |2 +
d
1
2
≤C
Ω
4
Ω
χ (p − 1) 2
2CD
Ω
upε+1−m |∇vε |2 +
Ω
u2ε |∇vε |2β−2
Ω
+ C for all t ∈ (0, Tmax ).
(3.6)
Lemma 3.4. There exists C > 0 such that the solution of (2.13) satisfies
|∇vε |q ≤ C for all t ∈ (0, Tmax )
Ω
(3.7)
with q ∈ [1, (N +Nr ). 2−r )+ Proof. Firstly, let us pick the above τ ∈ (0, Tmax ). Then by the regularity principle asserted by Lemma 2.5, we have ¯ ) with ∂vε (·,τ ) = 0 on ∂ Ω , so that in particular, we can pick K > 0 such that (uε (·, τ ), vε (·, τ )) ∈ C 2 (Ω ∂ν
∥uε (s)∥L∞ (Ω ) ≤ K ,
∥vε (s)∥L∞ (Ω ) ≤ K and ∥∇vε (s)∥L∞ (Ω ) ≤ K for all s ∈ [0, τ ].
(3.8)
Next, integrating the second equation in (2.13) with respect to space, we have
d dt
Ω
vε (x, t ) +
vε (x, t ) =
Ω
Ω
uε (x, t ),
(3.9)
which together with (3.7) implies that
Ω
vε (x, t ) ≤ λ + ( v0 (x) − λ)e−t for all t ∈ (0, Tmax ),
(3.10)
Ω
and choose some α > where λ is the same as Lemma 3.1. Now, we fix q < (N +Nr 2−r )+ q<
N 2
α−
1 2
1 2
+
+
N 2r
−
r −1 r
1 2
such that
,
(3.11)
which implies that
−α −
N
2
1 r
−
1
q
r r −1
> −1.
(3.12)
Now, involving the variation-of-constants formula for vε , we have
vε (t + τ ) = e−τ (A+Id ) vε (t ) +
t +τ
e−(t +τ −s)(A+Id ) uε (s)ds,
t ∈ (0, Tmax − τ ).
(3.13)
t
Hence, it follows from (2.3), (3.10), (3.13) and the Young’s inequality that
∥(A + Id )α vε (t + τ )∥Lq (Ω ) t +τ −α− N2 1− 1q −α− N2 1r − 1q −µ(t +τ −s) ≤ C (q) e ∥uε (s)∥Lr (Ω ) ds + c τ ∥vε ∥L1 (Ω ) (t + τ − s) t
≤ C (q)
t +τ
t
+ cτ
−α− N2
≤ C (q)
−α− N2
1 1 r −q
r r −1
− rr−µ1 (t +τ −s)
(t + τ − s) e 1− 1q λ+ v0 (x) − λ e−t
r −r 1
1r
t +τ
∥uε (s)∥
ds t
r Lr (Ω ) ds
Ω
+∞
σ 0
−α− N2
1 1 r −q
r r −1
− rr−µ1 σ
e
dσ
r −r 1
1
λ r + cτ
−α− N2 1− 1q
λ+ v0 (x) − λ e−t . Ω
(3.14)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
Hence, due to (3.12), (3.14) and (2.2), we have
Ω
|∇vε (t + τ )|q ≤ C for all t ∈ (0, Tmax − τ )
(3.15)
for all q ∈ [1, (N +Nr ). Finally, in view of (3.8) and (3.15), we can get (3.7). 2−r )+ Lemma 3.5. Assume that m ≥ 1, κ ≥ max{ N2 ,
max m + 1, 2 −
2 N
−m
γ 2
}, β > 1 and γ < max{ NN−1 , (N +Nr }. If 2−r )+
N (2m − 2)κ − N + 2κ N 2 (2κ − γ )+
(2N β − N γ + 2γ ) + 2 −
2 N
− m,
then for all small δ > 0, we can find a constant C := C (p, β, γ , δ) > 0 such that
Ω
uεp+1−m |∇vε |2 ≤ δ
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C for all t ∈ (0, Tmax ). L (Ω )
(3.16)
Ω
Proof. By the Hölder inequality, we have
J1 :=
uεp+1−m |∇vε |2
Ω
′ uκε (p+1−m)
≤ Ω
κ1 Ω
p+m−1 2(p+1−m) 2 p+m−1 2κ ′ (p+1−m) = uε p+m−1 L
(Ω )
|∇vε |
2κ
1′ κ
∥∇vε ∥2L2κ (Ω ) ,
(3.17)
where κ and κ ′ satisfy κ1′ + κ1 = 1. Since, m ≥ 1, κ ≥ 2
κ
p+1−m
≤
p+m−1
p+m−1κ −1
≤
N N −2
N 2
and p ≥ 1 + m, we have
,
which together with Lemma 2.2 implies that
p+m−1 2(p+1−m) 2 p+m−1 uε 2κ ′ (p+1−m) p+m−1 L
(Ω )
p+m2 −1 µ1 ≤ C4 ∇ uε 2
L (Ω )
p+m−1 1−µ1 2 uε p+m2 −1 L
(Ω )
p+m−1 2 + uε
2(pp++m1−−1m) 2
L p+m−1 (Ω )
p+m−1 2(p+1−m)µ1 p+m−1 2 ≤ C5 + 1 ∇ uε
(3.18)
L2 (Ω )
with some positive constants C4 , C5 and
κ1 =
N [p+m−1] 2
1−
N 2
N (p+m−1) 2κ ′ (p+1−m)
− +
N [p+m−1] 2
= [p + m − 1]
N 2
−
1−
N 2
N 2κ ′ (p+1−m)
+
N [p+m−1] 2
∈ (0, 1). γ
On the other hand, due to Lemma 2.2, Lemma 3.2 and Lemma 3.4 and the fact that β ≤ 2βκ ≤
2N , N −2
we have
2
∥∇vε ∥2L2κ (Ω ) = ∥ |∇vε |β ∥ β2κ
L β (Ω )
2κ2 2(1−κ2 ) 2 ≤ C6 ∥∇|∇vε |β ∥L2β(Ω ) ∥ |∇vε |β ∥ γ β + ∥|∇vε |β ∥ βγ L β (Ω )
2κ2 β β ≤ C7 ∥∇|∇vε | ∥L2 (Ω ) + 1 , with some positive constants C6 , C7 and Nβ
κ2 =
γ
1−
− N 2
Nβ 2κ
+
Nβ
γ
N
=β
γ
1−
− N 2
N 2κ
+
Nβ
γ
∈ (0, 1).
L β (Ω )
(3.19)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2611
Inserting (3.18)–(3.19) into (3.17) and using the Young’s inequality, we have
p+m−1 2(p+1−m)κ1 2κ2 p+m−1 2 ∥∇|∇vε |β ∥ 2β + 1 J1 ≤ C8 ∇ u + 1 ε 2 L (Ω ) L (Ω )
1 1 N γ2 − κ p+m−1 N (p+1−m− κ ′ ) N Nβ 1− N + N [p+m−1] β 1− 2 + γ 2 2 2 + 1 = C8 + 1 ∇ uε 2 ∥∇|∇vε | ∥L2 (Ω ) L (Ω )
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C9 for all t ∈ (0, Tmax ). ≤δ L (Ω )
(3.20)
Ω
Here we have use the fact that p <
N (2m−2)κ−N +2κ N 2 (2κ−γ )+
(2N β − N γ + 2γ ) + 2 −
2 N
− m and p > 2 −
γ
2 N
− m.
Nβ
Lemma 3.6. Assume that β > 1, γ < max{ NN−1 , (N +Nr } and [γ −2(β−1)]+ ≥ θ ≥ 2β−2+N . If 2−r )+
p > max 1, 2 −
2
2θ (N − 1)
− m, 1 − m +
N
N
,
2θ − 1 2θ γ + 2N θ − N γ
(2N β − N γ + 2γ ) + 2 −
2 N
−m ,
then for all small δ > 0, we can find a constant C := C (p, θ , γ , β, δ) > 0 such that
Ω
u2ε |∇vε |2β−2 ≤ δ
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C for all t ∈ (0, Tmax ). L (Ω )
(3.21)
Ω
Proof. Firstly, due to the Hölder inequality, we have
J2 :=
Ω
u2ε |∇vε |2β−2
≤ Ω
u2ε θ
θ1 Ω
p+m−1 4 2 p+m−1 = uε p+4mθ−1 L
|∇vε |(2β−2)θ
′
1′ θ
(2β−2)
(Ω )
∥∇vε ∥ (2β−2)θ ′ L
(Ω )
,
(3.22)
where θ and θ ′ satisfy θ1 + θ1′ = 1. 2θ(N −1) On the other hand, with the help of p > 1 − m + and Lemma 2.2, we conclude that N
p+m−1 4 2 p+m−1 uε p+4mθ−1 L
(Ω )
p+m2 −1 κ3 ≤ C10 ∇ uε 2
p+m−1 (1−κ3 ) 2 uε p+m2 −1
L (Ω )
L
(Ω )
p+m−1 2 + uε
p+m4 −1 2
L p+m−1 (Ω )
p+m−1 4κ3 p+m−1 2 ≤ C11 + 1 2 ∇ uε
(3.23)
L (Ω )
with some positive constants C10 , C11 and
κ3 =
N [p+m−1] 2 N 1 2
−
N (p+m−1) 4θ N [p+m−1] 2
−
+
= [p + m − 1]
1−
N 2 N 2
N 4θ N [p+m−1] 2
− +
∈ (0, 1).
γ Nβ On the other hand, it then follows from 2(β−1) ≤ θ ′ ≤ (N −2)(β−1) , Lemma 2.2, Lemma 3.2 and Lemma 3.4 that (2β−2)
∥∇vε ∥ (2β−2)θ ′ L
(Ω )
= ∥ |∇vε |β ∥
2β−2
L
β (2β−2)θ ′ β (Ω )
(2β−2)κ4 (2β−2)(1−κ4 ) (2β−2) ≤ C12 ∥∇|∇vε |β ∥L2 (Ωβ) ∥ |∇vε |β ∥ γ β + ∥ |∇vε |β ∥ γ β L β (Ω )
(2β−2)κ4 β
≤ C13 ∥∇|∇vε |β ∥L2 (Ω )
L β (Ω )
+1
(3.24)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
with some positive constants C12 , C13 and Nβ
γ
κ4 =
Nβ (2β−2)θ ′ N + Nγβ 2
−
1−
N
=β
γ
−
1−
N
(2β−2)θ ′ N + Nγβ 2
∈ (0, 1).
θ−1 Inserting (3.23)–(3.24) into (3.22) and using p > 2θγ +22N (2N β − N γ + 2γ ) + 2 − θ−N γ
1
2β−2
1
2 N
− m and Lemma 2.2, we have
N γ − θ′ p+m−1 N 2− θ Nβ 1− N + N [p+m−1] 1− N + γ 2 β 2 2 2 + 1 J2 ≤ C14 + 1 2 ∇ uε ∥∇|∇vε | ∥L2 (Ω )
L (Ω )
m+p−1 2 ∇ uε 2 + δ∥∇|∇vε |β ∥22 + C15 for all t ∈ (0, Tmax ). ≤δ L (Ω )
(3.25)
Ω
Lemma 3.7. Assume that m satisfies (1.6). Then there exists number β¯ > 2, such that
∥uε (·, t )∥Lp (Ω ) + ∥∇vε (·, t )∥L2β (Ω ) ≤ C for all p ≥ 1 and β > β¯ . Proof. The proof of this lemma can be divided into three parts. Case r < NN+2 : by m > 2 − N2 , there exists 1 ≤ l0 < NN−1 such that
(N + l0 )(m − 1) − (N − l0 ) > 0. Nl (m−1)−(N −l )
Thus, choosing β¯ ≥ max{3N , (N +0l )(m−1)−(N −0 l ) }, γ = l0 , κ = 0 0
max 1, 2 −
2
− m, 1 − m +
N
2θ (N − 1)
N
N 2
and θ =
2
= max 1, 2 − ≤
N
β 2
, then
− m, 1 − m +
β(N − 1)
N
β −1 2 (2N β − Nl0 + 2l0 ) + 2 − − m β l0 + N β − Nl0 N
(3.26)
and N (2m − 2)κ − N + 2κ N2
(2κ − l0
)+
(2N β − Nl0 + 2l0 ) + 2 −
for all β ≥ β¯ . Also due to m > 2 −
2 N
2 N
−m=
m−1 N − l0
(2N β − Nl0 + 2l0 ) + 2 −
2 N
− m.
(3.27)
Nl (m−1)−(N −l )
and β¯ ≥ max{3N , (N +0l )(m−1)−(N −0 l ) }, we have 0 0
β −1 2 m−1 2 (2N β − Nl0 + 2l0 ) + 2 − − m < (2N β − Nl0 + 2l0 ) + 2 − − m β l0 + N β − Nl0 N N − l0 N
(3.28)
for all β ≥ β¯ . Now, choosing δ small enough in Lemmas 3.5–3.6, then the solution of (2.13) satisfies that d dt
Ω
upε +
Ω
1 |∇vε |2β + r upε + |∇vε |2β ≤ C26 , 2
Ω
(3.29)
Ω
which implies that d dt
y(t ) + C27 y(t ) ≤ C28 ,
where y := Ω upε + Ω |∇vε |2β . Thus a standard ODE comparison argument implies boundedness of y(t ) for all t ∈ (0, Tmax ). Clearly, ∥uε (·, t )∥Lp (Ω ) and ∥∇vε (·, t )∥L2β (Ω ) are bounded for all t ∈ (0, Tmax ). Finally, with the help of the Hölder inequality, we can get the results. 2 Case N + ≥ r ≥ NN+2 : by m > 2 − N2r+2 , there exists NN−1 ≤ q0 < N +Nr2−r such that 2
(N + q0 )(m − 1) − (N − q0 ) > 0. Nq (m−1)−(N −q )
Thus, choosing β¯ ≥ max{3N , (N +q0 )(m−1)−(N −0q ) }, γ = q0 , κ = 0 0
max 1, 2 −
2 N
− m, 1 − m +
2θ (N − 1) N
N 2
≥
= max 1, 2 −
q0 2
2 N
and θ =
β 2
, then
− m, 1 − m +
β(N − 1)
N
β −1 2 ≤ (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N
(3.30)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
2613
and N (2m − 2)κ − N + 2κ N2
(2κ − q0
)+
(2N β − Nq0 + 2q0 ) + 2 −
2 N
−m=
m−1 N − q0
(2N β − Nq0 + 2q0 ) + 2 −
2 N
−m
(3.31)
Nq (m−1)−(N −q )
for all β ≥ β¯ . Also due to m > 2 − N2r and β¯ ≥ max{3N , (N +q0 )(m−1)−(N −0q ) }, we have +2 0 0
β −1 2 m−1 2 (2N β − Nq0 + 2q0 ) + 2 − − m < (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N N − q0 N for all β ≥ β¯ . Employing the same arguments as in the proof of case r < Case r > q0 ≥
N +2 : 2
N 2
There exists N −1 ≤ q0 < N
Nr (N +2−r )+
N +2 , N
(3.32)
we conclude the results.
such that
.
Thus, choosing β¯ ≥ 3N, γ = q0 , κ =
max 1, 2 −
2 N
− m, 1 − m +
and θ =
β
2θ (N − 1)
q0 2
2
N
, then obviously,
2 β(N − 1) = max 1, 2 − − m, 1 − m + N
N
β −1 2 ≤ (2N β − Nq0 + 2q0 ) + 2 − − m β q0 + N β − Nq0 N N (2m − 2)κ − N + 2κ 2 < (2N β − Nq0 + 2q0 ) + 2 − − m 2 + N (2κ − q0 ) N = +∞ for all β ≥ β¯ and m ≥ 1. Invoking the same arguments as in the proof of case r < of Lemma 3.7 is complete.
N +2 , N
(3.33)
we conclude the results. The proof
Underlying the estimates established above, we can derive the following boundedness results by invoking a Moser-type iteration (see Lemma A.1 in [6]) and standard parabolic regularity arguments. Lemma 3.8. Let (1.6) hold. Then one can find (εj )j∈N ⊂ (0, 1) such that εj → 0 as j → ∞ and that
∥uε (·, t )∥L∞ (Ω ) ≤ C for all t ∈ (0, Tmax )
(3.34)
∥vε (·, t )∥W 1,∞ (Ω ) ≤ C for all t ∈ (0, Tmax ).
(3.35)
and
By virtue of (2.15) and Lemma 3.8, the local-in-time solution can be extended to the global-in-time solution. Lemma 3.9. Assume that m satisfies (1.6). Then one can find C > 0 independent of ε ∈ (0, 1) such that
∥uε (·, t )∥L∞ (Ω ) ≤ C for all t ∈ (0, ∞)
(3.36)
∥vε (·, t )∥W 1,∞ (Ω ) ≤ C for all t ∈ (0, ∞).
(3.37)
and
As a straightforward result of Lemma 3.9, the following lemma gives uniform Hölder regularity properties of vε , ∇vε and uε . Here we only state the lemma, as for the detailed proof, readers can refer to the arguments of Lemma 3.18 and Lemma 3.19 in [22]. Lemma 3.10. Suppose that (1.6) is valid. Then one can find ζ ∈ (0, 1) such that for some C > 0
∥uε (·, t )∥
C
ζ ζ, 2
(Ω ×[t ,t +1])
≤ C for all t ∈ (0, ∞)
(3.38)
and such that for any τ > 0 there exists C (τ ) > 0 fulfilling
∥∇vε (·, t )∥
C
ζ ζ, 2
(Ω ×[t ,t +1])
≤ C for all t ∈ (τ , ∞).
(3.39)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
4. Regularity properties of time derivatives With the help of the estimates established before, we can obtain the boundedness property of the time derivatives of certain powers of uε on a fixed finite time interval, which will contribute to passing to the limit in the first equation in (2.13). Employing almost exactly the same arguments as in the proof of Lemmas 3.22–3.23 in [22] we can conclude the following lemmas. Since the procedure is rather standard, we may confine ourselves to an outline. Lemma 4.1. Assume that m satisfies (1.6) and p > 1 satisfies that p ≥ m − 1. Then for any fixed T > 0, there exists C (T ) > 0 with the property that T
p+m−1 2 ∇ uε 2 ≤ C (T )
(4.1)
Ω
0
for all ε ∈ (0, 1). Proof. Firstly, with the help of Lemma 3.10, for all ε ∈ (0, 1), we can fix a positive constant C1 such that and |∇vε | ≤ C1 in Ω × (0, ∞).
uε ≤ C 1
(4.2)
Let ς > m, ς ≥ 2(m − 1) and p := ς − m + 1. Then with the help of ς > m and ς ≥ 2(m − 1), we derive that p > 1 and p ≥ m − 1. Now, taking upε−1 as the test function for the first equation of (2.13) and using the Young’s inequality, we obtain 1 d p dt
∥uε ∥pLp (Ω ) + (p − 1)
Ω
upε−2 Dε (uε )|∇ uε |2 ≤ χ (p − 1)
Ω
≤ χ (p − 1) ≤
p−1
Ω
CD (p − 1)
uε
2
+µ
upε−1 ∇ uε · ∇vε + µ
Ω
p
uε −
Ω
|∇ uε | |∇vε | + µ
uεm+p−3 |∇ uε |2 +
p+r −1
uε
Ω
Ω
upε −
p
Ω
uεp+r −1
Ω
uε −
χ 2 (p − 1)
2CD
uεp+r −1
Ω
Ω
upε+1−m |∇vε |2
,
(4.3)
which implies that 1 d p dt
p uε Lp (Ω )
∥ ∥
+
CD (p − 1)
2
Ω
+p−3 um |∇ uε |2 ≤ ε
χ 2 (p−1)
Ω
2CD
uεp+1−m |∇vε |2 + µ
Ω
upε −
Ω
uεp+r −1 .
(4.4)
Now, in view of (4.2), we integrate (4.4) with respect to t over (0, T ) for some fixed T > 0 and then have 1
p
Ω
upε (·, T ) +
CD (p − 1)
T
2
0
Ω
+p−3 um |∇ uε |2 ≤ ε
≤
χ 2 (p − 1) 2CD
χ 2 (p − 1) 2CD
T
0
p
Ω
p+3−m
C1
uεp+1−m |∇vε |2 + µC1 T |Ω | + p
T |Ω | + µC1 T |Ω | +
1 p
p
Ω
u0 .
1 p
p
Ω
u0 (4.5)
On the other hand, by p = ς − m + 1, we have T
Ω
0
2 uς− |∇ uε |2 = ε
T
0
Ω
+p−3 um |∇ uε |2 ≤ C2 (1 + T ) ε
for some positive constant C2 . The proof of Lemma 4.1 is completed.
(4.6)
Lemma 4.2. Let (1.6) hold. Then one can find ε ∈ (0, 1) such that for some C > 0
∥∂t uε (·, t )∥(W 2,2 (Ω ))∗ ≤ C for all t ∈ (0, ∞).
(4.7)
0
Moreover, let ς > m and ς ≥ 2(m − 1). Then for all T > 0 and ε ∈ (0, 1) there exists C (T ) > 0 such that T
0
∥∂t uςε (·, t )∥(W lN ,2 (Ω ))∗ dt ≤ C (T ) for all t ∈ (0, ∞) and lN > 0
N +2 2
and l ∈ N.
(4.8)
Proof. Firstly, with the help of (4.2) and Dε ≤ D + 2ε (for all ε ∈ (0, 1)), we have Dε (uε ) ≤ C3
in Ω × (0, ∞) for all ε ∈ (0, 1),
where C3 := ∥D∥L∞ ((0,C1 )) + 2.
(4.9)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
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Now, testing the first equation by certain ϕ ∈ C0∞ (Ω ), we have
Ω
uε,t (·, t )ϕ =
Ω
= Ω
where Hε (s) :=
s
[∇ · (Dε (uε )∇ uε ) − χ∇ · (uε ∇vε )] ϕ +
Hε (uε )1ϕ + χ
uε ∇vε · ∇ϕ + µ
Ω
Ω
Ω
µuε − urε ϕ
uε ϕ −
Ω
urε ϕ
for all t ∈ (0, ∞),
(4.10)
Dε (τ )dτ for s ≥ 0. On the other hand, by Dε ≤ D + 2ε and uε ≥ 0, we also derive
0
Hε (uε ) ≤ C3 := C1 (∥D∥L∞ ((0,C1 )) + 2)
in Ω × (0, ∞) for all ε ∈ (0, 1).
(4.11)
Therefore, by (4.2) and (4.9)–(4.11), we obtain
r −1 2 uε,t (·, t )ϕ ≤ C3 |1ϕ| + C1 χ |∇ϕ| + C1 (µ + C1 ) |ϕ| Ω
Ω
Ω
(4.12)
Ω
for all t ∈ (0, ∞) and for all ε ∈ (0, 1). Thus, (4.12) yields (4.7). 1 Now, for any fixed ψ ∈ C0∞ (Ω ), multiplying the first equation by uς− ψ , we have ε 1
ς
Ω
∂t uςε (·, t )ψ =
Ω
uςε −1 [∇ · (Dε (uε )∇ uε ) − χ∇ · (uε ∇vε )] ψ +
= −(ς − 1)
Ω
+ χ (ς − 1) +µ
Ω
uςε −2 Dε (uε )|∇ uε |2 ψ
uςε ψ
− Ω
Ω
− Ω
r −1 uς+ ψ ε
1 uς− µuε − urε ψ ε
Ω
uςε −1 Dε (uε )∇ uε
uςε −1 ∇ uε · ∇vε ψ + χ
Ω
· ∇ψ
uςε ∇vε · ∇ψ
for all t ∈ (0, ∞).
(4.13)
Next, we will estimate the right-hand sides of (4.13). To this end, assuming that p := ς − m + 1, then ς > m and ς ≥ 2(m − 1) yield to p > 1 and p ≥ m − 1. Now, with the help of (4.2), we also derive that
µ
Ω
uςε ψ −
ς
Ω
uςε +r −1 ψ ≤ C1 |Ω |(µ + C1r −1 )∥ψ∥L∞ (Ω )
for all ε ∈ (0, 1).
(4.14)
Moreover, by (4.1), (4.13)–(4.14) and the Young’s inequality and applying the same arguments as in the proof of Lemma 3.22 in [22] we conclude that there exists C4 > 0 such that
ς−2 2 ∂t uς (·, t )ψ ≤ C4 u |∇ u | + 1 ∥ψ∥W 1,∞ (Ω ) . ε ε ε Ω
(4.15)
Ω
We choose lN ∈ N large enough to satisfy lN > exists C5 > 0 such that
∥∂t uςε (·, t )∥(W lN ,2 (Ω ))∗ ≤ C5
0
N +2 2
l ,2
and hence W0N (Ω ) ↩→ W 1,∞ (Ω ). Hence, (4.15) implies that there
Ω
uςε −2 |∇ uε |2 + 1
Now, combining (4.6) and (4.16), we can get (4.8).
for all t ∈ (0, ∞) and any ε ∈ (0, 1).
(4.16)
Lemma 4.3. Assume that m satisfies (1.6) Then there exists (εj )j∈N ⊂ (0, 1) such that εj → 0 as j → ∞ and that uε → u
a.e. in Ω × (0, ∞),
uε ⇀ u
weakly star in L (Ω × (0, ∞)), 2 ,2 W0
in
0 Cloc
([0, ∞); (
vε → v in
0 Cloc
¯ × [0, ∞)), (Ω
uε → u
(4.17)
∞
(Ω )) ), ∗
(4.18) (4.19) (4.20)
¯ × [0, ∞)) (Ω
(4.21)
∇vε → ∇v in L∞ (Ω × (0, ∞))
(4.22)
∇vε → ∇v in
0 Cloc
and
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
with some triple (u, v) which is a global weak solution of (2.13) in the sense of Definition 2.1. Moreover, u satisfies 0 u ∈ Cω−∗ ([0, ∞); L∞ (Ω )).
(4.23)
Proof. Firstly, by the same arguments as in the proof of Lemma 4.1 in [22] we can get that (4.19)–(4.23). On the other hand, by Lemma 3.9, we derive that for certain u ∈ L∞ (Ω × (0, ∞)), (4.18) holds. Moreover, we next fix ς > m satisfying ς ≥ 2(m − 1) and set p := 2ζ − m + 1, then by (4.6), we have for each T > 0, (uςε )ε∈(0,1) is bounded in L2 ((0, T ); W 1,2 (Ω )). Furthermore, Lemma 4.2 implies that
(∂t uςε )ε∈(0,1)
l ,2
is bounded in L1 ((0, T ); (W0N (Ω ))∗ ) for each T > 0 and lN >
N +2 2
.
Hence, by Aubin–Lions lemma (Theorem III.2.3 of [23]), we also derive that the strong precompactness of (uςε )ε∈(0,1) in L2 (Ω × (0, T )). Thus, we can pick a suitable subsequence such that uςε → z ς for some nonnegative measurable z : Ω × (0, Ω ) → R. In view of (4.18) and the Egorov theorem, we have z = u necessarily, so that (4.17) is valid. Now, we will prove that (u, v) is a global weak solution of (2.13) in the sense of Definition 2.1. To this end, by (4.17), (4.18), (4.20)–(4.22) we have (2.9)–(2.10) hold and u and v are nonnegative functions. Finally, letting ε := εj ↘ 0 and using (4.17)–(4.22), we can get (2.11)–(2.12). The proof of Lemma 4.3 is completed. Now we can easily prove Theorem 2.1. Proof of Theorem 2.1. Combining Lemma 3.9 with Lemma 4.3, we prove Theorem 2.1. 5. Asymptotic behavior In this section we study the long-time behavior for (2.13) in the case µ = 0. As the first step, we give the decay property separately for the integrals of the solution components u and v . Lemma 5.1. Let µ = 0 and m satisfy (1.6). Then we have
u( x , t ) ≤
Ω
u0 (x)
Ω
1−r
1−1 r + (r − 1)|Ω |
1−r
for all t ∈ (0, ∞).
t
(5.1)
Proof. Let t > 0 and s ∈ (0, t ). Since µ = 0, it follows from an integration by parts to the first equation in (2.13) and the Hölder inequality that d
ds
Ω
u(x, s) = −
u (x, s) ≤ −|Ω |
1 −r
r
Ω
Ω
u(x, s)
r
for all s ∈ (0, t ),
(5.2)
which implies that
Ω
1−r 1−r u(x, t ) − u0 (x) ≥ (r − 1)|Ω |1−r t .
Hence, (5.1) holds.
(5.3)
Ω
As a consequence, we obtain a basic decay property also for the second solution component. Lemma 5.2. There exists C > 0 such that
C
v(x, t ) ≤
for all t ∈ (0, Tmax ). (5.4) 1 (1 + t ) r −1 Proof. Firstly, let z (t ) := Ω v(x, t ), t ∈ [0, Tmax ). Next, integrating the second equation in (2.13) we see that there exists C1 > 0 such that ′ z (t ) = −z (t ) + u(x, t ), Ω
Ω
1
≤ −z (t ) + C1 (1 + t ) 1−r for all t ∈ (0, Tmax ). 1 min{1, r − } 1 max{ We now let C2 := 2 v (x), 2} and define Ω 0 1 − min{1, r − } 1
z¯ (t ) := C2 (t + 2)
for all t ≥ 0.
(5.5)
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
Then z¯ (0) =
C2 min{1, 1 } r −1
≥
2
Ω
2617
v0 (x) = z0 and 1
1
− min 1,
1
−1
− min 1,
1
1
r −1 r −1 + C2 (t + 2) − C1 (1 + t ) 1−r (t + 2) r −1 1 1 1 −1 1 − min 1, r − 1 ≥ C2 (t + 2) − min 1, 2 t +2 r −1 1 1 1 1 − min 1, r − min 1 , r− 1 1 1 − r C2 − 2C1 (t + 2) (1 + t ) + (t + 2) 2 1 1 − min 1, r − −1 1 1 ≥ C2 (t + 2) − 2 2 1 1 1 1 1 − min 1, r − 1+min 1, r − min 1, r − + 1− 1 1 1 r + (t + 2) C2 − C1 2 (t + 1)
z¯ ′ (t ) + z¯ (t ) − C1 (1 + t ) 1−r = −C2 min 1,
2
≥ 0 for all t > 0.
(5.6)
By comparison, we thus infer that z (t ) ≤ z¯ (t ) for all t ∈ (0, Tmax ), which directly establishes (5.4).
Now thanks to the precompactness of both (u(·, t ))t >1 and (v(·, t ))t >1 implied by Lemmas 4.1 and 4.2, the latter two decay properties readily entail the first of our main results on large time behavior, addressing the case when µ = 0. Lemma 5.3. Let µ = 0 and m satisfy (1.6). Then with (u, v) as given by Theorem 2.1, we have ∗
u(·, t ) ⇀ 0 in L∞ (Ω ) as t → ∞.
(5.7)
Proof. We suppose that (5.7) is not valid. Then there exists a sequence (tk )k∈N such that tk → ∞ as k → ∞, and that for ˜ ∈ L1 ( Ω ) certain ψ
Ω
˜ ≥ C1 u(x, tk )ψ
for all k ∈ N
(5.8)
with some C1 > 0. From Lemma 3.9, we can find C2 > 0 such that u(x, t ) ≤ C2
for a.e. (x, t ) ∈ Ω × (0, ∞).
(5.9)
˜ L1 (Ω ) ≤ Due to the density of C0∞ (Ω ) in L1 (Ω ), we can pick a ψ ∈ C0∞ (Ω ) such that ∥ψ − ψ∥ that Ω
u(x, tk )ψ ≥
≥
Ω
C1 , 2C2
thus it follows from (5.9)
˜ − ∥u(·, tk )∥L∞ (Ω ) ∥ψ − ψ∥ ˜ L1 (Ω ) u(x, tk )ψ
C1
for all k ∈ N .
2
(5.10)
Now, in light of (4.7), there exists C3 > 0 fulfilling
∥uε (·, t ) − uε (·, s)∥(W 2,2 (Ω ))∗ ≤ C3 |t − s| for all t , s ∈ (0, ∞) and ε ∈ (0, 1),
(5.11)
0
which along with convergence (4.19) yields
∥u(·, t ) − u(·, s)∥(W 2,2 (Ω ))∗ ≤ C3 |t − s| for all t , s ∈ (0, ∞).
(5.12)
0
If we let τ ∈ (0, 1) such that
τ≤
C1 4C3 ∥ψ∥W 2,2 (Ω )
,
0
then for all t ∈ (tk , tk + τ ) with k ∈ N we have
u(x, tk )ψ − ≤ ∥u(·, t ) − u(·, t )∥ 2,2 ∗ ∥ψ∥ 2,2 u ( x , t )ψ (W0 (Ω )) W0 (Ω ) Ω
Ω
≤ C3 |tk − t |∥ψ∥W 2,2 (Ω ) 0
≤
C1 4
for all t ∈ (tk , tk + τ ) with k ∈ N
(5.13)
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J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
which combined with (5.10) entails
Ω
C1
u(x, t )ψ ≥
4
for all t ∈ (tk , tk + τ ) with k ∈ N .
(5.14)
We set uk (x, s) := u(x, tk + s), (x, s) ∈ Ω × (0, τ ) for all k ∈ N . Then an application of (5.14) to uk yields
Ω
C1
uk (x, s)ψ ≥
4
for all k ∈ N and s ∈ (0, τ ).
(5.15)
Additionally, from Lemma 5.1, we can find C4 > 0 satisfying
Ω
uk ( x , s ) =
Ω
u( x , t k + s ) ≤
C4 1
(1 + tk + s) r −1
for all k ∈ N .
(5.16)
Case r = 2: we integrate (5.15) with respect to s over (0, τ ) and then obtain C1 τ 4
τ
≤ 0 τ Ω ≤ Ω
0
≤ C4 ln
uk (x, s)ψ(x)dxds C4 ψ(x) 1 + tk + s
1 + tk + τ 1 + tk
dxds
Ω
ψ(x)dx → 0 as k → ∞,
(5.17)
which is absurd and thereby proves that virtually (5.7) is valid. Case r ̸= 2: integrate (5.15) with respect to s over (0, τ ) and then obtain C1 τ 4
τ
≤ 0 τ Ω ≤
uk (x, s)ψ(x)dxds C4 ψ(x)
dxds
1
(1 + tk + s) r −1 2−r 2−r 1−r ψ(x)dx → 0 as k → ∞, ≤ C4 (1 + tk + τ ) 1−r − (1 + tk ) 1−r 2−r Ω 0
Ω
which is absurd and thereby proves that virtually (5.7) is valid.
(5.18)
With the help of Lemmas 3.10 and 5.2, we can achieve the decay property of v : Lemma 5.4. Let µ = 0 and m satisfy (1.6). Then with (u, v) as given by Theorem 2.1, we have
v(·, t ) → 0 in L∞ (Ω ) as t → ∞.
(5.19)
Proof. Assume that (5.19) does not hold, then we can find (tk )k∈N and C1 > 0 such that tk → ∞ as k → ∞ and
∥v(·, tk )∥L∞ (Ω ) ≥ C1 for all k ∈ N .
(5.20)
On the basis of Lemma 3.10, an application of the Arzela–Ascoli theorem allows for an extraction of a subsequence from v(·, tk )k∈N such that
v(·, tk ) → v∞ in L∞ (Ω ) as k → ∞
(5.21)
¯ ), where we still denote the subsequence by v(·, tk )k∈N . However, Lemma 5.2 shows with a nonnegative function v∞ ∈ C (Ω that v(·, t ) → 0 in L1 (Ω ) as t → ∞, thereby necessarily v∞ ≡ 0, which is contradiction to (5.20) and thus (5.19) is valid. 0
Proof of Theorem 2.2. The conclusions of Theorem 2.2 follow from Lemmas 5.3 and 5.4. Acknowledgments The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work. The first author is partially supported by the National Natural Science Foundation of China (No. 11601215) and partially supported by the Doctor Start-up Funding of Ludong University (No. LA2016006). The second author is partially supported by NNSFC (No. 11571363).
J. Zheng, Y. Wang / Computers and Mathematics with Applications 72 (2016) 2604–2619
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