Global existence of solutions to a two dimensional attraction–repulsion chemotaxis system in the attractive dominant case with critical mass

Nonlinear Analysis 190 (2020) 111615

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Nonlinear Analysis www.elsevier.com/locate/na

Global existence of solutions to a two dimensional attraction–repulsion chemotaxis system in the attractive dominant case with critical mass Toshitaka Nagai a , Tetsuya Yamada b ,∗ a

Department of Mathematics, Hiroshima University, Higashihiroshima, 739-8526, Japan Course of General Education (Science), National Institute of Technology, Fukui College, Sabae, Fukui 916-8507, Japan b

article

info

abstract

Article history: Received 4 April 2019 Accepted 25 August 2019 Communicated by Enrico Valdinoci MSC: 35B45 35K15 35K55

We study the Cauchy problem for an attraction–repulsion chemotaxis system in two dimensions: ∂t u = ∆u − ∇ · (u∇(β1 v1 − β2 v2 )), 0 = ∆v1 − λ1 v1 + u, 0 = ∆v2 − λ2 v2 + u with nonnegative initial data u0 and positive constants β1 , β2 , λ1 , λ2 . The purpose of this paper is to show that∫the Cauchy problem admits the global nonnegative solutions in time if (β1 − β2 ) 2 u0 dx = 8π in the attractive R dominant case β1 > β2 . © 2019 Elsevier Ltd. All rights reserved.

Keywords: Attraction–repulsion chemotaxis system a priori estimate Global existence

1. Introduction We are concerned with the following system ⎧ ∂t u = ∆u − ∇ · (u∇(β1 v1 − β2 v2 )), ⎪ ⎪ ⎪ ⎨0 = ∆v − λ v + u, 1 1 1 ⎪ 0 = ∆v − λ 2 2 v2 + u, ⎪ ⎪ ⎩ u(0, x) = u0 (x),

t > 0, x ∈ R2 , t > 0, x ∈ R2 , t > 0, x ∈ R2 , x ∈ R2 ,

(P)

where βi and λi (i = 1, 2) are all positive constants and the initial data u0 is always assumed that u0 ≥ 0 on R2 , u0 ̸≡ 0 and u0 ∈ L1 (R2 ) ∩ L∞ (R2 ). ∗ Corresponding author. E-mail addresses: [email protected] (T. Nagai), [email protected] (T. Yamada).

https://doi.org/10.1016/j.na.2019.111615 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

(1.1)

2

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Here Lp (R2 ) is the usual Lebesgue space on R2 with the norm ∥ · ∥Lp for 1 ≤ p ≤ ∞, and in what follows, the following abbreviations are used: Lp = Lp (R2 ),

∥ · ∥ p = ∥ · ∥ Lp .

The system (P) is a simplified mathematical model which is introduced in [10] to describe the aggregation of Microglia in the central nervous system. In the system (P), the functions u, v1 and v2 denote the density of Microglia, the concentration of attractive and repulsive chemical substances, respectively. The system (P) in the case β2 = 0 becomes a minimal version of the original Keller–Segel model (see [7,8]): ⎧ 2 ⎪ ⎨∂t u = ∆u − β1 ∇ · (u∇v1 ), t > 0, x ∈ R , 2 (KS) 0 = ∆v1 − λ1 v1 + u, t > 0, x ∈ R , ⎪ ⎩ 2 u(0, x) = u0 (x), x∈R , where λ1 is nonnegative. The mass conservation for the solution u to (KS) holds and plays an important role in the existence of nonnegative global solutions to (KS). Indeed, in the case β1 ∥u0 ∥1 ≤ 8π, there exist nonnegative solutions globally in time (see, e.g., [2–4,11–13,17]), meanwhile, in the case β1 ∥u0 ∥1 > 8π, nonnegative solutions may blow up in finite time (see, e.g., [1,3,9,17]). For the Cauchy problem (P), Shi–Wang [15] established that if the initial data u0 satisfies (1.1), then there exists uniquely a solution (u, v1 , v2 ) on [0, T ] × R2 for some 0 < T < ∞ with the following properties: (i) (u, v1 , v2 ) is smooth on (0, T ]×R2 and solves the Cauchy problem (P) in the classical sense on (0, T ]×R2 , (ii) u, v1 and v2 are positive on (0, T ] × R2 , (iii) for all 0 ≤ t ≤ T , ∥u(t)∥1 = λ1 ∥v1 (t)∥1 = λ2 ∥v2 (t)∥1 = ∥u0 ∥1 . (1.2) Furthermore, it was proven in [15] that the nonnegative solutions exist globally in time and decay to zero as t → ∞ in the repulsive dominant case β1 < β2 , and also shown that there exists a nonnegative solution blowing up in finite time if (β1 − β2 )∥u0 ∥1 > 8π in the attractive dominant case β1 > β2 . Nagai–Yamada [14] established the boundedness of nonnegative solutions in the balance case β1 = β2 and also showed that the Cauchy problem (P) has nonnegative global in time solutions if (β1 − β2 )∥u0 ∥1 < 8π in the attractive dominant case β1 > β2 . However, to the best of our knowledge, the global in time solvability of nonnegative solutions to the Cauchy problem (P) remains still open when β1 > β2 and (β1 − β2 )∥u0 ∥1 = 8π. The aim of this paper is to show the existence of nonnegative global in time solutions to the Cauchy problem (P) under the condition (β1 − β2 )∥u0 ∥1 = 8π in the attractive dominant case β1 > β2 . Our main theorem reads as follows: Theorem 1.1. Let β1 > β2 . Then, under the condition (β1 − β2 )∥u0 ∥1 = 8π, the Cauchy problem (P) has the nonnegative global in time solution (u, v1 , v2 ). The proof of Theorem 1.1 relies on an a priori estimate for the modified entropy u(t)) dx and the following proposition.

∫

R2

(1 + u(t)) log(1 +

Proposition 1.2 ([14, Proposition 1.2.]). Let Tmax be the maximal existence time of the nonnegative solution (u, v1 , v2 ) to the Cauchy problem (P). If Tmax is finite, then ∫ lim sup (1 + u(t)) log(1 + u(t)) dx = ∞. (1.3) t↑Tmax

R2

Hence, for all 1 < p ≤ ∞, lim sup ∥u(t)∥p = ∞. t↑Tmax

(1.4)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

3

To establish the a priori estimate for the modified entropy in the subcritical mass case (β1 − β2 )∥u0 ∥1 < 8π, in [14] we introduced the following functional ∫ ∫ 1 u(t)(β1 v1 (t) − β2 v2 (t)) dx, (1.5) Fm [u(t)] = (1 + u(t)) log(1 + u(t)) dx − 2 R2 R2 ∫ and applied the Brezis–Merle type inequality on R2 [14, Lemma 2.6] to control the essential term R2 u(t) v1 (t) dx in the second integral term on the right-hand side of (1.5). However, in the critical mass case (β1 − β2 )∥u0 ∥1 = 8π, it does not seem that the Brezis–Merle type inequality on R2 is applicable. Instead of that, we employ methods used in [13] to get the a priori estimate. We first show the a priori bound for the modified entropy on exterior domains {x; |x| > R} (see Theorem 3.1) thanks to the uniform smallness ∫ of |x|>R u(t) dx with respect to t ∈ [0, T ] for large R, and next for the modified entropy on interior domains {x; |x| < R} (see Proposition 7.6) by applying the Brezis–Merle inequality on bounded domains (see Corollary 2.7). As a result, it enables us to establish the a priori estimate for the modified entropy ∫ (1 + u(t)) log(1 + u(t)) dx. R2 The rest of this paper is organized as follows. In Section 2, we collect useful tools needed in the proof of Theorem 1.1. Section 3 is devoted to giving the a priori estimate for the modified entropy in the exterior domain. In Sections 4 and 5, we get L2 , L3 and L∞ estimates for the solutions in the exterior domain. In Section 6, we prove an estimate for the functional Hint (t; R) (see (6.2) for the definition). In Section 7, we show the a priori estimate for the modified entropy in the interior domain and then give the proof of Theorem 1.1. Before closing this section, we introduce the following notation. First of all, we denote by Z+ the set of all nonnegative integers and put |α| = α1 + α2 for α = (α1 , α2 ) ∈ Z2+ . We indicate by ∂tm and ∂jm any partial derivative of order m with respect to variables t and xj ∈ R, respectively, and set ∇ = t (∂1 , ∂2 ) and ∂xα = ∂1α1 ∂2α2 for α = (α1 , α2 ) ∈ Z2+ . For an open set Ω of R2 , Lp (Ω ) is the Lebesgue space on Ω with the norm ∥ · ∥Lp (Ω) for 1 ≤ p ≤ ∞ and W k,p (Ω ) the Sobolev space for k ∈ N and 1 ≤ p ≤ ∞. Also, for a Banach space X and T > 0, we denote by Lp ((0, T ); X) the set of all X-valued Lebesgue functions on an interval (0, T ) with the norm ∥ · ∥Lp (0,T ;X) for 1 ≤ p ≤ ∞. We use a universal constant C to denote a various constant, and C(∗, . . . , ∗) when C depends on the quantities appearing in parentheses. 2. Preliminaries In this section we prepare some lemmas used in the proof of Theorem 1.1. First of all, we begin with the estimates of the Bessel kernel Bλ (λ > 0) defined by ∫ ∞ Bλ (x) = e−λσ G(σ, x) dσ, x ∈ R2 , (2.1) 0 2

where G = G(t, x) is the heat kernel, namely, G(t, x) = (4πt)−1 e−|x|

/(4t)

.

Lemma 2.1 ([6,16]). The Bessel kernel Bλ (λ > 0) given by (2.1) satisfies that ∥∂xα Bλ ∥p < ∞ for 1 ≤ p ≤ ∞ if |α| = 0 and 1 ≤ p < 2 if |α| = 1. For λ > 0 and f ∈ Lp (1 ≤ p ≤ ∞), the function (λ − ∆)−1 f represents the convolution of the Bessel kernel Bλ and f , that is, (λ − ∆)−1 f = Bλ ∗ f. This function on R2 for f ∈ Lp (1 < p < ∞) is in W 2,p (R2 ) and a solution of (λ − ∆)v = f in R2 . Then Young’s inequality for convolution and Lemma 2.1 yield the Lp estimates on (λ − ∆)−1 f in Lemma 2.2 below, which are used in the proof of Theorem 1.1.

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

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Lemma 2.2 ([14, Lemma 2.1]). For λ > 0, it holds that ∥(λ − ∆)−1 f ∥p ≤ C(λ, p, q)∥f ∥q , 1 ≤ q ≤ p < ∞,

(2.2)

∥(λ − ∆)−1 f ∥∞ ≤ C(λ, q)∥f ∥q , 1 < q ≤ ∞,

(2.3)

∥∇(λ − ∆)−1 f ∥p ≤ C(λ, p)∥f ∥1 , 1 ≤ p < 2,

(2.4)

∥∇(λ − ∆)−1 f ∥p ≤ C(λ, p, q)∥f ∥q , q ≤ p < 2q/(2 − q), 1 < q < 2,

(2.5)

∥∇(λ − ∆)−1 f ∥∞ ≤ C(λ, q)∥f ∥q , 2 < q ≤ ∞.

(2.6)

For f ∈ Lp (1 ≤ p ≤ ∞), the heat semigroup et∆ is defined by ∫ G(t, x − y)f (y) dy, t > 0, x ∈ R2 , (et∆ f )(x) = R2

which is a solution of the heat equation. As one of properties for the heat semigroup, the following Lp –Lq estimates are well-known (see, e.g., [5]): Let 1 ≤ q ≤ p ≤ ∞, m ∈ Z+ and α ∈ Z2+ . Then, ∥∂tm ∂xα et∆ f ∥p ≤ Ct−(1/q−1/p)−m−|α|/2 ∥f ∥q

for f ∈ Lq .

(2.7)

Applying the Lp –Lq estimates (2.7) for et∆ , we have the following lemma. Lemma 2.3.

Let q ∈ [1, ∞] and T ∈ (0, ∞). For f ∈ L∞ (0, T ; Lq ), set ∫ F (t) =

t

e(t−s)∆ f (s) ds,

0 < t < T.

0

Then the following assertions hold: (i) For all 1 ≤ p < ∞, ∥F (t)∥p ≤ C(p)t1/p ∥f ∥L∞ (0,T ;L1 ) , 0 < t < T.

(2.8)

(ii) For all 1 < q ≤ ∞ and q ≤ p ≤ ∞, ∥F (t)∥p ≤ C(p, q)t1−1/q+1/p ∥f ∥L∞ (0,T ;Lq ) , 0 < t < T.

(2.9)

(iii) For all 1 ≤ q < 2 and q ≤ p < 2q/(2 − q), ∥∇F (t)∥p ≤ C(p, q)t1/2−1/q+1/p ∥f ∥L∞ (0,T ;Lq ) , 0 < t < T.

(2.10)

∥∇F (t)∥p ≤ C(p)t1/p ∥f ∥L∞ (0,T ;L2 ) , 0 < t < T.

(2.11)

(iv) For all 2 ≤ p < ∞,

(v) For all 2 < q ≤ ∞ and q ≤ p ≤ ∞, ∥∇F (t)∥p ≤ C(p, q)t1/2−1/q+1/p ∥f ∥L∞ (0,T ;Lq ) , 0 < t < T.

(2.12)

The following function inequalities are helpful in getting a priori estimates for the solutions to the Cauchy problem (P).

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

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Lemma 2.4 ([13, Lemma 2.2]). Assume that ϕ ∈ W 1,∞ satisfies 0 ≤ ϕ ≤ 1 and ∇ϕ1/2 ∈ L∞ . Then, for all nonnegative functions f ∈ W 1,2 ∩ L1 , the following inequalities hold: ) (∫ ) (∫ ∫ 2 |∇f | 2 ϕ dx f ϕ dx ≤2 f dx R2 {f >1}∩ supp ϕ {f >1} 1 + f (∫ )2 (∫ ) + f |∇ϕ1/2 | dx + 4 f ϕ dx , (2.13) R2 R2 (∫ )∫ ∫ 2 f 3 ϕ dx ≤ε (1 + f ) log(1 + f ) dx |∇f | ϕ dx R2

R2

supp ϕ

(∫ +

)2 ∫ 3/2 1/2 f |∇ϕ | dx + C(ε)

R2

f ϕ dx,

(2.14)

R2

where ε > 0 and C(ε) → ∞ as ε → 0. Lemma 2.5 ([13, Lemma 2.3]). Let Ω be a measurable set in R2 and f a nonnegative measurable function satisfying f , f log(2 + |x|) ∈ L1 (Ω ). Then the conditions f log f ∈ L1 (Ω ) and (1 + f ) log(1 + f ) ∈ L1 (Ω ) are equivalent. Furthermore, ∫ ∫ ∫ (1 + f ) log(1 + f ) dx ≤ 2 f | log f | dx + (2 log 2) f dx, (2.15) Ω Ω ∫Ω ∫ ∫ f | log f | dx ≤ (1 + f ) log(1 + f ) dx + 2α f log(2 + |x|) dx Ω Ω Ω ∫ 1 1 dx, (2.16) + e Ω (2 + |x|)α where 2 < α < ∞. The following lemma is about the Brezis–Merle type inequality shown in [13]. Lemma 2.6 ([13, Lemma 2.7]). Let Ω be a bounded smooth domain in R2 . For g ∈ L2 (Ω ), let v ∈ W 2,2 (Ω ) be a solution of −∆v = g inΩ . If ∥g∥L1 (Ω) < 4π, then { } ∫ 4π 2 e|v(x)| dx ≤ d(Ω )2 exp sup |v(x)| , 4π − ∥g∥L1 (Ω) x∈∂Ω Ω where d(Ω ) is the diameter of Ω . Using Lemma 2.6, we obtain the following Corollary, which is a crucial key to give an a priori estimate for the modified entropy in interior domains. Corollary 2.7. Let Ω be a bounded smooth domain in R2 . For λ > 0 and g ∈ L2 (Ω ), let w ∈ W 2,2 (Ω ) be a solution of −∆w + λw = g in Ω . If ∥g∥L1 (Ω) < 4π, then { } ∫ 4π 2 2 |w(x)| e dx ≤ d(Ω ) exp sup |w(x)| , 4π − ∥g∥L1 (Ω) x∈∂Ω Ω where d(Ω ) is the diameter of Ω . Proof . Let v be in W 2,2 (Ω ) satisfying −∆v = |g| in Ω , Then v ≥ 0 in Ω .

v = |w| on ∂Ω .

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

6

Let wj (j = 1, 2) be in W 2,2 (Ω ) satisfying − ∆w1 + λw1 = g+ in Ω ,

w1 = w+ on ∂Ω ,

− ∆w2 + λw2 = g− in Ω ,

w2 = w− on ∂Ω ,

where g+ = max{g, 0} and g− = max{−g, 0}. Then w = w1 − w2 and wj ≥ 0 in Ω for each j = 1, 2. As g± ≤ |g| and w± ≤ |w|, we have −∆(v − wj ) = (|g| − g± ) + λwj ≥ 0 in Ω ,

v − w± ≥ 0 on ∂Ω ,

and hence, by the maximum principle, wj ≤ v in Ω

(j = 1, 2),

from which it follows that |w| ≤ v in Ω . As a consequence, Corollary 2.7 follows from Lemma 2.6.

□

3. Modified entropy estimates in exterior domains Let T ∈ (0, ∞) and (u, v1 , v2 ) be the nonnegative solution to the Cauchy problem (P) on [0, T ) × R2 . Taking u(t0 ) as the initial data for some t0 ∈ (0, T ), in addition to (1.1) we may assume that u0 > 0 (x ∈ R2 ), u0 ∈ Lp (1 ≤ p ≤ ∞), u0 ∈ W k,p (k ∈ N, 1 < p ≤ ∞), because u is positive on (0, T ) × R2 and has sufficient regularity properties. Also, putting ψ = β1 v1 − β2 v2 and β = β1 − β2 , we remark that (u, ψ) satisfies ∂t u =∆u − ∇ · (u∇ψ),

(3.1)

− ∆ψ =βu − λ1 β1 v1 + λ2 β2 v2 .

(3.2)

As vj = (λj − ∆)−1 u (j = 1, 2) and ∥u(t)∥1 = ∥u0 ∥1 (t > 0), applying Lemma 2.2 as f = u(t), we observe that for j = 1, 2 and t > 0, ∥vj (t)∥p ≤ C∥u0 ∥1

(1 ≤ p < ∞),

∥∇vj (t)∥p ≤ C∥u0 ∥1

(3.3)

(1 ≤ p < 2).

(3.4)

It is apparent that (3.3) and (3.4) are valid for ψ = β1 v1 − β2 v2 . The aim of this section is to obtain an a priori estimate for the modified entropy in the exterior domain {x; |x| > R}. Theorem 3.1.

There exists a sufficiently large constant R0 > 1 depending only on ∥u0 ∥1 and T such that ∫ sup (1 + u(t)) log(1 + u(t)) dx ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ), (3.5) 0
∫

T

0

2

|x|≥R0

0

∫

|x|≥R0

∫

T

∫

|∇u| dxdt ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ), 1+u

(3.6)

u2 dxdt ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ).

(3.7)

|x|≥R0

To prove Theorem 3.1, we introduce a cut-off function ϕR ∈ C0∞ (R2 ) with the following properties: There exists a positive constant C independent of R such that for all R > 1, 0 ≤ ϕR ≤ 1 (x ∈ R2 ), ϕR (x) = 0 (|x| ≤ R/2), ϕR (x) = 1 (|x| ≥ R), 5/6

1/2

1/3

2/3

|∇ϕR | ≤ CR−1 ϕR , |∇ϕR | ≤ CR−1 ϕR , |∂xα ϕR | ≤ CR−2 ϕR (|α| = 2). Firstly we begin with the following two lemmas.

(3.8)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Let R be a constant with R > 1. Then it holds that ∫ ∫ u(t)ϕR dx ≤ u0 ϕR dx + CR−2 (∥u0 ∥1 + ∥u0 ∥21 )T,

Lemma 3.2.

R2

0 < t < T.

7

(3.9)

R2

Proof . Multiplying (3.1) by the cut-off function ϕR and integrating by parts, we obtain ∫ ∫ ∫ d u(t)ϕR dx = u(t)∆ϕR dx + u(t)⟨∇ψ(t), ∇ϕR ⟩ dx. dt R2 R2 R2 Noting that vj = (λj − ∆)−1 u and ∇Bλj (x − y) = −∇Bλj (y − x) (x, y ∈ R2 , x ̸= y, j = 1, 2), we have 2

∫

1∑ (−1)j−1 βj u(t)⟨∇ψ(t), ∇ϕR ⟩ dx = 2 j=1 R2

∫∫ u(t, x)u(t, y)Aj (x, y; R) dxdy, R2 ×R2

where ⟨ ⟩ Aj (x, y; R) := ∇Bλj (x − y), ∇ϕR (x) − ∇ϕR (y) . We then get |Aj (x, y; R)| ≤ CR−2 by using |∂i ∂j ϕR | ≤ CR−2 (i, j = 1, 2), |∇Bλj (x)| ≤ C/|x| (j = 1, 2) and applying the mean value theorem. Hence, by ∥u(t)∥1 = ∥u0 ∥1 , ∫ u(t)⟨∇ψ(t), ∇ϕR ⟩ dx ≤ CR−2 ∥u0 ∥21 . R2

Remarking that

∫

R2

u(t)∆ϕR dx ≤ CR−2 ∥u0 ∥1 due to |∆ϕR | ≤ CR−2 , we have d dt

∫ R2

u(t)ϕR dx ≤ CR−2 (∥u0 ∥1 + ∥u0 ∥21 ).

Therefore, integrating the inequality just above from 0 and t with respect to the time variable, we obtain (3.9). □ Lemma 3.3.

For all 0 < t < T , it holds that d H(t; R) + dt

∫ R2

2

5

∑ |∇u(t)| ϕR dx = Ik (t; R), 1 + u(t) k=1

where ∫ H(t; R) =

{(1 + u(t)) log(1 + u(t)) − u(t)}ϕR dx, R∫2

I1 (t; R) = β {u2 (t) − u(t) log(1 + u(t))}ϕR dx (β = β1 − β2 ), 2 ∫ R I2 (t; R) = {(1 + u(t)) log(1 + u(t)) − u(t)}∆ϕR dx, 2 R∫ I3 (t; R) = − log(1 + u(t))ψ(t)⟨∇u(t), ∇ϕR ⟩ dx, 2 ∫R I4 (t; R) = − {(1 + u(t)) log(1 + u(t)) − u(t)}ψ(t)∆ϕR dx, ∫ R2 I5 (t; R) = {u(t) − log(1 + u(t))}(λ2 β2 v2 (t) − λ1 β1 v1 (t))ϕR dx. R2

(3.10)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

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Proof . As in the proof of [13, Proposition 3.2], ) } |∇u|2 ϕR (1 + u) log(1 + u) − u ϕR + 1+u { ( ) } =∇ · ∇u log(1 + u) ϕR − (1 + u) log(1 + u) − u ∇ϕR − u log(1 + u) ∇ψϕR ( ) + (1 + u) log(1 + u) − u ∆ϕR + u log(1 + u)⟨∇ψ, ∇ϕR ⟩

∂t

{(

(3.11)

+ u⟨∇ log(1 + u), ∇ψ⟩ϕR , and the sum of the last two terms on the right-hand side of (3.11) is rewritten as u log(1 + u)⟨∇ψ, ∇ϕR ⟩ + u⟨∇ log(1 + u), ∇ψ⟩ϕR {( ) ( ) } = ∇ · u − log(1 + u) ∇ψϕR + (1 + u) log(1 + u) − u ψ∇ϕR ( ) − log(1 + u) ψ⟨∇u, ∇ϕR ⟩ − (1 + u) log(1 + u) − u ψ∆ϕR ( ) − u − log(1 + u) ∆ψϕR .

(3.12)

Substituting −∆ψ = βu − λ1 β1 v1 + λ2 β2 v2 in the last term on the right-hand side of (3.12) and plugging the resulting equality into (3.11), we obtain (3.10) by integrating (3.11) on R2 . □ Proof of Theorem 3.1. By the argument similar to that in [13, Proposition 3.2], the sum of the first four terms on the right hand side of (3.10) is estimated as follows: )∫ ( ∫ 4 2 ∑ |∇u(t)| 1 +C u(t) dx ϕR dx + J 1 (t; R), Ij (t; R) ≤ 4 1 + u(t) 2 |x|≥R/2 R j=1 where J 1 (t; R) :=C∥u(t)ϕR ∥1 + CR−2 (∥u0 ∥1 + ∥u0 ∥21 ) + CR−4 ∥ψ(t)∥66 + CR−4 ∥ψ(t)∥33 + CR−4 . For the term I5 (t; R), by u − log(1 + u) ≥ 0 (u ≥ 0) and vj ≥ 0 (j = 1, 2), we get ∫ I5 (t; R) ≤ λ2 β2 u(t)v2 (t)ϕR dx. R2

Using H¨ older’s inequality and Young’s inequality and applying (2.13) as f = u(t) and ϕ = ϕR , we have that (∫ )1/2 (∫ )1/2 I5 (t; R) ≤ λ2 β2 u2 (t)ϕR dx v22 (t)ϕR dx R2 R2 ∫ ∫ ≤ u2 (t)ϕR dx + C v22 (t)ϕR dx 2 2 R R (∫ )∫ 2 |∇u(t)| ≤2 u(t) dx ϕR dx + J 2 (t; R), 1 + u(t) 2 |x|≥R/2 R where J 2 (t; R) := 4∥u(t)ϕR ∥1 + CR−2 ∥u0 ∥1 + C∥v2 (t)∥22 . Therefore, summing up the estimates above, we obtain that ( )∫ ∫ 5 2 ∑ 1 |∇u(t)| Ij (t; R) ≤ +C u(t) dx ϕR dx + J(t; R), 4 |x|≥R/2 R2 1 + u(t) j=1

(3.13)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

9

where J(t; R) := J 1 (t; R) + J 2 (t; R). Since ∥vj (t)∥p ≤ C∥u0 ∥1 (1 ≤ p < ∞) by (3.3), J(t; R) is estimated as J(t; R) ≤ C(1 + R−2 )(∥u0 ∥1 + ∥u0 ∥21 ) + CR−4 (1 + ∥u0 ∥31 + ∥u0 ∥61 ). (3.14) ∫ ∫ We now use (3.9) to estimate |x|>R/2 u(t) dx. As R2 u0 ϕR dx → 0 (R → ∞) in (3.9), we can take R0 depending only on ∥u0 ∥1 and T such that ∫ u(t) dx ≤ 1/4 for all R ≥ R0 , 0 < t < T. |x|≥R/2

Then, by (3.13) and (3.14), 5 ∑

Ij (t; R) ≤

j=1

1 2

∫ R2

2

|∇u(t)| ϕR dx + C(∥u0 ∥1 , T ). 1 + u(t)

(3.15)

Substituting (3.15) on the right-hand side of (3.10), and then integrating the resulting inequality from 0 to t, we see that for all R ≥ R0 and t ∈ (0, T ), ∫ ∫ 2 1 t |∇u| H(t; R) + ϕR dxds ≤ H(0; R) + C(∥u0 ∥1 , T ). (3.16) 2 0 R2 1 + u Hence, noting that H(0; R0 ) ≤ ∥u0 ∥22 by log(1 + u) ≤ u (u ≥ 0), we observe from (3.16) with R = R0 that for all 0 < t < T , ∫ (1 + u(t)) log(1 + u(t)) dx ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ), |x|≥R0

∫ t∫ |x|≥R0

0

2

|∇u| dxds ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ). 1+u

As a result, we get (3.5) and (3.6). Also (3.7) follows from (3.6) by applying (2.13) as f = u(t) and ϕ = ϕR0 . □ 4. L2 and L3 estimates in exterior domains In this section, L2 and L3 estimates for u in the exterior domain {x; |x| > R} are given. In what follows, the functions ψ and ϕR and the constant R0 > 1 are the same as in Section 3. Firstly we begin with the following lemma. Lemma 4.1.

Let p ≥ 2. Then it holds that for all 0 < t < T , 1 d p dt

4(p − 1) u (t)ϕR dx + p2 R2

∫

p

∫ R2

2

|∇up/2 (t)| ϕR dx =

4 ∑

Kpj (t; R),

j=1

where ∫ p−1 β up+1 (t)ϕR dx (β = β1 − β2 ), = p 2 R ∫ 1 up (t)(1 − ψ(t))∆ϕR dx, Kp2 (t; R) = p R2 ∫ 3 Kp (t; R) = − up−1 (t)ψ(t)⟨∇u(t), ∇ϕR ⟩ dx, R2 ∫ p−1 Kp4 (t; R) = − up (t)(λ1 β1 v1 (t) − λ2 β2 v2 (t))ϕR dx. p R2 Kp1 (t; R)

(4.1)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

10

Proof . As in the proofs of [13, Lemmas 3.4 and 3.6], differentiating (3.2) and integrating by parts, we have the desired equality (4.1). □

∫

R2

up (t)ϕR dx in t, using (3.1) and

The following lemma gives us the L2 estimate for u in the exterior domain. Lemma 4.2.

For all R ≥ 2R0 , it holds that ∫ ∫ ∫ 1 T 2 2 |∇u| dxdt ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ). sup u (t) dx + 2 0 |x|≥R 0
Proof . From (4.1) with p = 2 we have ∫ ∫ 4 ∑ 1 d 2 2 u (t)ϕR dx + |∇u(t)| ϕR dx = K2j (t; R). 2 dt R2 2 R j=1

(4.2)

(4.3)

As in the proof of [13, Lemma 3.4], by H¨ older’s inequality and Young’s inequality, the sum of the first three terms on the right hand side of (4.3) is estimated as follows: ∫ ∫ 3 ∑ 1 2 |∇u(t)| ϕR dx + L1 (t). (4.4) K2j (t; R) ≤ C u3 (t)ϕR dx + 2 2 2 R R j=1 Here, thanks to ∥vj (t)∥p ≤ C∥u0 ∥1 (1 ≤ p < ∞) by (3.3) and ψ = β1 v1 − β2 v2 , L1 (t) := CR0−4 (∥ψ(t)∥66 + ∥ψ(t)∥33 + 1) ≤ CR0−4 (∥u0 ∥61 + ∥u0 ∥31 + 1). For the term K24 (t; R), H¨ older’s inequality and Young’s inequality yield that (∫ )2/3 (∫ )1/3 1 3 K24 (t; R) ≤ u3 (t)ϕR dx |β1 λ1 v1 (t) − β2 λ2 v2 (t)| ϕR dx 2 2 R2 ∫ R ≤C u3 (t)ϕR dx + C∥β1 λ1 v1 (t) − β2 λ2 v2 (t)∥33 . R2

Therefore we have that ∫ ∫ ∫ d 2 u2 (t)ϕR dx + |∇u(t)| ϕR dx ≤ C u3 (t)ϕR dx + 2L2 (t). dt R2 R2 R2

(4.5)

Here, by the estimate of L1 (t) and (3.3), L2 (t) := L1 (t) + C∥β1 λ1 v1 (t) − β2 λ2 v2 (t)∥33 ≤ CR0−4 (∥u0 ∥61 + ∥u0 ∥31 + 1) + C∥u0 ∥31 . We now apply (2.14) as f = u(t) and ϕ = ϕR0 to obtain ∫ C u3 (t)ϕR dx R2 ( ∫ ≤ Cε

)∫ R2

|x|≥R/2

(∫ +C R2

2

|∇u(t)| ϕR dx

(1 + u(t)) log(1 + u(t)) dx

(4.6)

)2 1/2 u3/2 (t)|∇ϕR | dx + C(ε)∥u0 ∥1 , 1/2

where ε > 0 and C(ε) → ∞ (ε → 0). For the second term on the right-hand side of (4.6), by |∇ϕR | ≤ 1/3 CR−1 ϕR and ϕR (x) = 0 (|x| ≤ R/2), (∫ )2 1/2 C u3/2 (t)|∇ϕR | dx R2 (∫ )2 (4.7) ≤ CR−2

u3/2 (t) dx

|x|≥R/2

≤ CR0−2 ∥u0 ∥1 ∥u(t)∥2L2 (|x|≥R0 ) .

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Plugging (4.7) in (4.6) yields that for all t ∈ (0, T ) and R ≥ 2R0 , ∫ C u3 (t)ϕR dx R2( )∫ ∫ ≤Cε

2

|∇u(t)| ϕR dx

(1 + u(t)) log(1 + u(t)) dx

11

(4.8)

R2

|x|≥R/2

+ CR0−2 ∥u0 ∥1 ∥u(t)∥2L2 (|x|≥R0 ) + C(ε)∥u0 ∥1 . Thanks to (3.5), we can take ε > 0 such that for all R ≥ 2R0 , ∫ Cε sup (1 + u(t)) log(1 + u(t)) dx ≤ 1/2, 0≤t
|x|≥R/2

from which together with (4.8) it follows that for all t ∈ (0, T ) and R ≥ 2R0 , ∫ u3 (t)ϕR dx R2 ∫ 1 2 |∇u(t)| ϕR dx + CR0−2 ∥u0 ∥1 ∥u(t)∥2L2 (|x|≥R0 ) + C∥u0 ∥1 . ≤ 2 R2

(4.9)

Substituting (4.9) into (4.5) and then integrating the resulting inequality from 0 to t, we get ∫ ∫ ∫ ∫ 1 t 2 u2 (t)ϕR dx + |∇u| ϕR dxds ≤ u20 ϕR dx + L(t), 2 0 R2 R2 R2 where L(t) is given by t

∫

2

L(t) = 2

L (s) ds +

CR0−2 ∥u0 ∥1

0

∫ 0

t

∥u(s)∥2L2 (|x|≥R0 ) ds + C∥u0 ∥1 T.

Since L(t) ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ) by the estimate of L2 (t) and (3.7), we obtain (4.2). □ From Lemma 4.2, we have the following lemma. Lemma 4.3.

For all R ≥ 22 R0 , it holds that ∫ T∫ u4 dxdt ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ). 0

(4.10)

|x|≥R

Proof . By applying (2.13) as f = u2 (t) and ϕ = ϕR , and using Lemma 4.2, (4.10) is derived. □ The L3 estimate for u in the exterior domain is obtained as follows. Lemma 4.4.

For all R ≥ 23 R0 , it holds that ∫ ∫ T∫ sup u3 (t) dx + 0
|x|≥R

2

|∇u3/2 | dxdt ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ).

(4.11)

|x|≥R

0

Proof . (4.1) with p = 3 implies that 1 d 3 dt

∫ R2

u3 (t)ϕR dx +

8 9

∫ R2

2

|∇u3/2 (t)| ϕR dx =

4 ∑ j=1

K3j (t; R).

(4.12)

12

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

By remarking that 3u2 (t)∇u(t) = 2u3/2 (t)∇u3/2 (t), the term K32 (t; R) + K33 (t; R) is rewritten as ∫ ∫ 1 1 3 2 3 u (t)∆ϕR dx − u3 (t)ψ(t)∆ϕR dx K3 (t; R) + K3 (t; R) = 3 R2 3 R2 ∫ 2 u3/2 (t)ψ(t)⟨∇u3/2 (t), ∇ϕR ⟩ dx. − 3 R2 Therefore, by calculations similar to those in [13, Lemma 3.6], we can get the following estimate: 3 ∑

K3j (t; R) ≤ C∥u(t)∥4L4 (|x|≥R/2) + CR−2 ∥u(t)∥3L3 (|x|≥R/2) (4.13)

j=1

5 + 9

∫

3/2

|∇u

2

(t)| ϕR dx + CR

−8

R2

(∥ψ(t)∥44

+

∥ψ(t)∥88 ).

Also, by 0 ≤ ϕR ≤ 1, H¨ older’s inequality and Young’s inequality, the term K34 (t; R) is estimated as K34 (t; R) ∫ 2 u3 (t)(β1 λ1 v1 (t) − β2 λ2 v2 (t))ϕR dx =− 3 R2 (∫ )3/4 (∫ )1/4 2 4 4 ≤ u (t)ϕR dx |β1 λ1 v1 (t) − β2 λ2 v2 (t)| ϕR dx 3 R2 R2 2 ∑ ≤ C∥u(t)∥4L4 (|x|≥R/2) + C (βj λj )4 ∥vj (t)∥44 .

(4.14)

j=1

Thus, combining (4.13) and (4.14) with (4.12) and integrating the resulting inequality from 0 and t, we obtain that ∫ ∫ t∫ 2 u3 (t)ϕR dx + |∇u3/2 | ϕR dxds 2 2 R 0 ∫ ∫Rt 3 −2 ≤ u0 dx + CR ∥u(s)∥3L3 (|x|≥R/2) ds R2 0 ∫ t ∫ t (4.15) +C ∥u(s)∥4L4 (|x|≥R/2) ds + CR−8 (∥ψ(s)∥44 + ∥ψ(s)∥88 ) ds 0

+C

2 ∑

0

(βj λj )4

j=1

∫

t

∥vj (s)∥44 ds.

0

∥u(t)∥3L3 (|x|≥R/2)

Noting that ≤ ∥u0 ∥1 + ∥u(t)∥4L4 (|x|≥R/2) and ∥vj (t)∥p ≤ C∥u0 ∥1 (1 ≤ p < ∞) by (3.3), and using (4.10), it follows from (4.15) that for all t ∈ (0, T ), ∫ ∫ t∫ 2 3 u (t)ϕR dx + |∇u3/2 | ϕR dxds ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ), R2

0

R2

which implies the desired inequality (4.11).

□

5. L∞ estimates in exterior domains The aim of this section is to give the L∞ estimates for ∂xα u (|α| ≤ 2) in the exterior domain. 5.1. L∞ estimates for u in exterior domains Firstly, we begin with the following two lemmas.

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Lemma 5.1.

13

For all R ≥ 22 R0 and each j = 1, 2, it holds that sup ∥vj (t)ϕR ∥∞ ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ),

(5.1)

0
sup ∥vj (t)∥L∞ (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ),

(5.2)

0
sup ∥ψ(t)∥L∞ (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ).

(5.3)

0
Proof . From the second and third equations of (P), the functions vj ϕR (j = 1, 2) satisfy (λj − ∆)(vj ϕR ) =λj vj ϕR − ∆(vj ϕR ) =λj vj ϕR − (λj vj − u)ϕR − 2⟨∇vj , ∇ϕR ⟩ − vj ∆ϕR =uϕR − 2⟨∇vj , ∇ϕR ⟩ − vj ∆ϕR

(5.4)

= : hj . For the L3/2 norm of hj (j = 1, 2), we have ∥hj (t)∥3/2 ≤∥u(t)ϕR ∥3/2 + 2∥⟨∇vj (t), ∇ϕR ⟩∥3/2 + ∥vj (t)∆ϕR ∥3/2 ≤∥u(t)ϕR ∥3/2 + 2∥∇vj (t)∥3/2 ∥∇ϕR ∥∞ + ∥vj (t)∥3/2 ∥∆ϕR ∥∞ . By (4.2), we have that for all t ∈ (0, T ) and R ≥ 22 R0 , (∫ ∥u(t)ϕR ∥3/2 ≤ 0
( ≤

3/2 u(t)ϕR

dx + u(t)≥1

)2/3 3/2 u2 (t)ϕR

dx (5.6)

)2/3

∫

2

∥u0 ∥1 + sup 0
∫

u (t) dx

(5.5)

≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ).

|x|≥R/2

Using (3.3) and (3.4), we have that for all t ∈ (0, T ) and R ≥ 22 R0 , 2∥∇vj (t)∥3/2 ∥∇ϕR ∥∞ + ∥vj (t)∥3/2 ∥∆ϕR ∥∞ ≤ C(R−1 + R−2 )∥u0 ∥1 .

(5.7)

Hence, combining (5.6) and (5.7) with (5.5), we see that for all R ≥ 22 R0 , sup ∥hj (t)∥3/2 ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T )

(j = 1, 2).

(5.8)

0
As vj ϕR = (λj − ∆)−1 hj (j = 1, 2), applying (2.3) as f = hj (t) implies (5.1). Also (5.2) follows from (5.1) and |vj (t, x)| ≤ ∥vj (t)ϕR ∥L∞ (t ∈ (0, T ), |x| ≥ R), and (5.3) from (5.2) because of ψ = β1 v1 − β2 v2 . □ Lemma 5.2.

Let 3/2 ≤ p ≤ ∞. For all R ≥ 24 R0 and each j = 1, 2, the following estimates hold: sup ∥∇(vj (t)ϕR )∥p ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ),

(5.9)

0
sup ∥∇vj (t)∥Lp (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ),

(5.10)

0
sup ∥∇ψ(t)∥Lp (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ).

(5.11)

0
Proof . Let 3/2 ≤ p < 6 and j = 1, 2. By (5.4), (λj − ∆)(vj ϕR ) = hj , where hj = uϕR − 2⟨∇vj , ∇ϕR ⟩ − vj ∆ϕR , that is, vj (t)ϕR = (λj − ∆)−1 hj (t). Since ∥hj (t)∥3/2 ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ) by (5.8), applying (2.5), we have that for all R ≥ 22 R0 , sup ∥∇(vj (t)ϕR )∥p ≤ C sup ∥hj (t)∥3/2 ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ). 0
0
(5.12)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

14

We next estimate ∥hj (t)∥3 . By H¨ older’s inequality and (3.8), ∥hj (t)∥3 ≤∥u(t)∥L3 (|x|≥R/2) + CR−1 ∥∇vj (t)∥L3 (|x|≥R/2) + CR−2 ∥vj (t)∥3 .

(5.13)

Using (3.3) and (4.11), we have that for all R ≥ 24 R0 and all 0 < t < T , ∥u(t)∥L3 (|x|≥R/2) + CR−2 ∥vj (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ).

(5.14)

Since ∇(vj ϕR/2 ) = ∇vj (|x| ≥ R/2) due to ϕR/2 (x) = 1 and ∇ϕR/2 (x) = 0 (|x| ≥ R/2), it follows from (5.12) that for all R ≥ 23 R0 and all 0 < t < T , ∥∇vj (t)∥L3 (|x|≥R/2) = ∥∇(vj (t)ϕR/2 )∥L3 (|x|≥R/2) ≤ C(∥u0 ∥1 , ∥u0 ∥2 , T ).

(5.15)

Combining (5.14) and (5.15) with (5.13), we get R ≥ 2 4 R0 .

sup ∥hj (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ),

(5.16)

0
Hence, applying (2.6) as f = hj (t) yields that for all R ≥ 24 R0 , sup ∥∇(vj (t)ϕR )∥∞ ≤ C sup ∥hj (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ). 0
(5.17)

0
Therefore, by (5.12) and (5.17), we can obtain the desired estimate (5.9) for 3/2 ≤ p ≤ ∞. Also (5.10) follows from (5.9) and |∇vj (t, x)| = |∇(vj (t, x)ϕR (x))| (t > 0, |x| ≥ R), and (5.11) easily from (5.10) by ψ = β1 v1 − β2 v2 . □ We now use Lemma 5.2 to obtain the L∞ estimate for u in the exterior domain. Proposition 5.3.

For all R ≥ 25 R0 , it holds that sup ∥u(t)ϕR ∥∞ ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T ),

(5.18)

0
sup ∥u(t)∥L∞ (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T ).

(5.19)

0
Proof . Let R ≥ 25 R0 . From (3.1), the function uϕR satisfies ∂t (uϕR ) − ∆(uϕR ) = u(∆ϕR + ⟨∇ψ, ∇ϕR ⟩) + ∇ · (−u(ϕR ∇ψ + 2∇ϕR )) =: f1 + ∇ · f2 ,

(5.20)

2

0 < t < T, x ∈ R .

Therefore we can rewrite uϕR as u(t)ϕR = et∆ (u0 ϕR ) +

∫

t

e(t−s)∆ f1 (s) ds +

0 t∆

=: e

(u0 ϕR ) + F1 (t) + F2 (t),

∫

t

∇ · e(t−s)∆ f2 (s) ds

0

(5.21)

0 < t < T.

We first have ∥et∆ u0 ∥∞ ≤ C∥u0 ∥∞ (t > 0) by the L∞ − L∞ estimate (2.7) for et∆ . Since ∥u(t)∥L3 (|x|≥R) and ∥∇ψ(t)∥L∞ (|x|≥R) are bounded from above by a positive constant C(∥u0 ∥1 , ∥u0 ∥3 , T ) by (4.11) and (5.11), we have that for all 0 < t < T , ∥f1 (t)∥3 ≤ ∥u(t)∥L3 (|x|≥R/2) (∥∆ϕR ∥∞ + ∥∇ψ(t)∥L∞ (|x|≥R/2) ∥∇ϕR ∥∞ ) ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ),

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

15

and similarly, ∥f2 (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ). Hence, applying (2.9) yields that sup ∥F1 (t)∥∞ ≤ sup ∥f1 (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ). 0
0
Similarly, by (2.12), sup ∥F2 (t)∥∞ ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ). 0
Therefore, by the estimates above, we derive the desired estimate (5.18). Also, (5.19) follows from (5.18) by ϕR (x) = 1 (|x| ≥ R). □ 5.2. L∞ estimates for higher partial derivatives of u in exterior domains In this subsection, we are going to give L∞ estimates for ∂xα u (1 ≤ |α| ≤ 2) in the exterior domain. For this purpose, we prepare the following lemma. Lemma 5.4. For all R ≥ 26 R0 , it holds that ∫ ∫ 2 sup |∇u(t)| dx + 0
|x|≥R

T

∫

0

2

|∆u| dxdt |x|≥R

(5.22)

≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T ). Proof . As the lemma is proven by the method similar to that in [13, Lemma 3.9], we give only the outline of the proof. ∫ 2 Let R ≥ 26 R0 . Differentiating ( R2 |∇u| ϕR dx)/2 with respect to t and integrating by parts, we have ∫ ∫ ∫ 1 d 2 |∇u(t)| ϕR dx = − ∂t u(t)∆u(t)ϕR dx − ∂t u(t)⟨∇u(t), ∇ϕR ⟩ dx. 2 dt R2 R2 R2 By Eqs. (3.1) and (3.2) for u and ψ, we see that ∫ − ∂t u(t)∆u(t)ϕR dx 2 R∫ ∫ 2 =− |∆u(t)| ϕR dx + ⟨∇u(t), ∇ψ(t)⟩∆u(t)ϕR dx 2 2 R∫ ∫R −β u2 (t)∆u(t)ϕR dx + u(t)(λ1 β1 v1 (t) − λ2 β2 v2 (t))∆u(t)ϕR dx, R2

(5.23)

R2

where β = β1 − β2 . By H¨ older’s inequality and Young’s inequality, the first three terms on the right hand side of (5.23) are estimated as ∫ ∫ ∫ 2 − |∆u(t)| ϕR dx + ⟨∇u(t), ∇ψ(t)⟩∆u(t)ϕR dx − β u2 (t)∆u(t)ϕR dx 2 2 2 R ∫ R R 5 2 2 4 ≤− |∆u(t)| ϕR dx + 3β ∥u(t)∥L4 (|x|≥R/2) + 3∥∇ψ(t)∥2L∞ (|x|≥R/2) ∥∇u(t)∥2L2 (|x|≥R/2) . 6 R2 Similarly, ∫ u(t)(λ1 β1 v1 (t) − λ2 β2 v2 (t))∆u(t)ϕR dx ∫ 1 2 ≤ |∆u(t)| ϕR dx + 3∥λ1 β1 v1 (t) − λ2 β2 v2 (t)∥26 ∥u(t)∥3L3 (|x|≥R/2) . 12 R2 R2

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

16

Hence, by collecting these estimates, the following inequality holds: ∫ ∫ 3 2 |∆u(t)| ϕR dx + M 1 (t), − ∂t u(t)∆u(t)ϕR dx ≤ − 4 2 2 R R

(5.24)

where M 1 (t) :=3∥∇ψ(t)∥2L∞ (|x|≥R/2) ∥∇u(t)∥2L2 (|x|≥R/2) + 3β 2 ∥u(t)∥4L4 (|x|≥R/2) + 3∥λ1 β1 v1 (t) − λ2 β2 v2 (t)∥26 ∥u(t)∥3L3 (|x|≥R/2) . Next, as in the proof of [13, Lemma 3.9], ∫ − ∂t u(t)⟨∇u(t), ∇ϕR ⟩ dx 2 R∫ ∫ =− ∆u(t)⟨∇u(t), ∇ϕR ⟩ dx + ⟨∇u(t), ∇ψ(t)⟩⟨∇u(t), ∇ϕR ⟩ dx ∫ R2 ∫ R2 −β u2 (t)⟨∇u(t), ∇ϕR ⟩ dx + u(t){λ1 β1 v1 (t) − λ2 β2 v2 (t)}⟨∇u(t), ∇ϕR ⟩ dx. R2

R2

Analogously to the calculations for the terms on the right-hand side of (5.23), we have that ∫ ∫ 1 2 − ∂t u(t)⟨∇u(t), ∇ϕR ⟩ dx ≤ |∆u(t)| ϕR dx + M 2 (t), 4 R2 R2

(5.25)

where 1 1 M 2 (t) := ∥u(t)∥3L3 (|x|≥R/2) + ∥u(t)∥4L4 (|x|≥R/2) + C∥λ1 β1 v1 (t) − λ2 β2 v2 (t)∥66 3 4 + C(1 + ∥∇ψ(t)∥L∞ (|x|≥R/2) )∥∇u(t)∥2L2 (|x|≥R/2) . Hence, combining (5.24) and (5.25), we obtain ∫ ∫ d 2 2 |∇u(t)| ϕR dx + |∆u(t)| ϕR dx ≤ 2M (t), dt R2 R2

0 < t < T,

(5.26)

∫T where M (t) := M 1 (t) + M 2 (t). Because ∥vj (t)∥6 ≤ C∥u0 ∥1 due to (3.3), and 0 ∥∇u(t)∥2L2 (|x|≥2R ) dt, 0 ∥∇ψ(t)∥L∞ (|x|≥R/2) and ∥u(t)∥L∞ (|x|≥R/2) are bounded from above by a positive constant C(∥u0 ∥1 , ∥u0 ∥∞ , T ) by (4.2), (5.11) and (5.19), we have that for all 0 < t < T , ∫ T M (t) dt ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T ). 0

Therefore, integrating (5.26) from 0 to t with respect to the time variable yields that for all 0 < t < T , ∫ ∫ t∫ 2 2 |∇u(t)| ϕR dx + |∆u| ϕR dxds R2∫ 0 R2 (5.27) 2 ≤ |∇u0 | dx + C(∥u0 ∥1 , ∥u0 ∥∞ , T ). R2

Therefore, the desired inequality (5.22) is derived.

□

The following proposition is to show L∞ estimates for ∂xα u (1 ≤ |α| ≤ 2) in the exterior domain. Proposition 5.5.

For all R ≥ 27 R0 , the following estimates hold: sup ∥∇u(t)∥L∞ (|x|≥R) ≤ C(∥u0 ∥1 , ∥∇u0 ∥2 , ∥u0 ∥W 1,∞ , T ),

(5.28)

0
sup ∥∂xα u(t)∥L∞ (|x|≥R) ≤ C(∥u0 ∥1 , ∥∇u0 ∥2 , ∥u0 ∥W 2,∞ , T ),

0
|α| = 2.

(5.29)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

17

Proof . Following the method in [13, Lemma 3.9], we prove only (5.28), since (5.29) is done in a similar way (see the proof of [13, Lemma 3.11]). Let R ≥ 27 R0 . By (5.20) we recall that w = uϕR satisfies ∂t w = ∆w + g1 + g2 , 0 < t < T, x ∈ R2 , where g1 = − ⟨2∇ϕR + ϕR ∇ψ, ∇u⟩, g2 = − (∆ϕR )u + βϕR u2 − ϕR u(λ1 β1 v1 − λ2 β2 v2 ). Then we can rewrite w as w(t) = et∆ w(0) +

∫

t

e(t−s)∆ (g1 + g2 )(s) ds,

0 < t < T.

(5.30)

0

By (5.30), we observe that for all 0 < t < T , ∇w(t) = et∆ ∇w(0) +

∫

t

∇e(t−s)∆ (g1 + g2 )(s) ds.

0

Since ∥u(t)∥1 = ∥u0 ∥1 , and ∥vj (t)∥L∞ (|x|≥R/2) and ∥u(t)∥L∞ (|x|≥R/2) are bounded from above by a positive constant C(∥u0 ∥1 , ∥u0 ∥∞ , T ) by (5.2) and (5.19), taking into account the properties (3.8) of ϕR , we have that for all 1 ≤ p ≤ ∞, sup ∥g2 (t)∥p ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T ). (5.31) 0
As for g1 (t), since ∥∇ψ(t)∥L∞ (|x|≥R/2) is bounded from above by a positive constant C(∥u0 ∥1 , ∥u0 ∥3 , T ) by (5.11), we have that for p = 2, 3 and 0 < t < T , ( ) ∥g1 (t)∥p ≤ C + ∥∇ψ(t)∥L∞ (|x|≥R/2) ∥∇u(t)∥Lp (|x|≥R/2) (5.32) ≤ C(∥u0 ∥1 , ∥u0 ∥3 , T )∥∇u(t)∥Lp (|x|≥R/2) . Since ∥∇u(t)∥L2 (|x|≥R/2) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T ) by (5.22), by (5.31) and (5.32) for p = 2, we have sup ∥g1 (t) + g2 (t)∥2 ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T ). 0
Let 2 < p < ∞. Using the Lp –Lq estimate (2.7) for et∆ and applying (2.11) as f = g1 + g2 , we obtain that for all 0 < t < T , ∥∇w(t)∥p ≤ ∥∇w(0)∥p + CT 1/p sup ∥(g1 + g2 )(t)∥2 0
≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , ∥∇u0 ∥p , T ). Hence, as u(t, x) = w(t, x) (|x| ≥ R), sup ∥∇u(t)∥Lp (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , ∥∇u0 ∥p , T ).

(5.33)

0
Since ∥∇u(t)∥L3 (|x|≥R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , ∥∇u0 ∥3 , T ) (0 < t < T ) by (5.33) for p = 3, it follows from (5.31) and (5.32) for p = 3 that sup ∥g1 (t) + g2 (t)∥3 ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , ∥∇u0 ∥3 , T ). 0
Hence, applying (2.12) as f = g1 + g2 yields that for all 0 < t < T , ∥∇w(t)∥∞ ≤ ∥∇w(0)∥∞ + CT 1/6 sup ∥(g1 + g2 )(t)∥3 0
≤ C(∥u0 ∥1 , ∥∇u0 ∥2 , ∥u0 ∥W 1,∞ , T ). Therefore, we conclude (5.28) because of u(t, x) = w(t, x) (|x| ≥ R).

□

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

18

6. Some estimates in interior domains In this section, let φR ∈ C0∞ (R2 ) (R > 1) satisfy the following properties: { 1, |x| ≤ R, |∇φR | ≤ CR−1 , |∆φR | ≤ CR−2 , 0 ≤ φR ≤ 1, φR (x) = 0, |x| ≥ 2R, where C is a positive constant independent of R. We define Hint (t; R) by ∫ ∫ 1 Hint (t; R) := {u(t) log u(t) − u(t)}φR dx − u(t)ψ(t)φR dx, 2 R2 R2

(6.1)

(6.2)

where ψ = β1 v1 − β2 v2 . The purpose of this section is to give the following estimate for Hint (t; R). Proposition 6.1.

Let R ≥ 28 R0 . Then, it holds that for all 0 < t < T , ∫ t∫ 2 Hint (t; R) + u|∇(log u − ψ)| φR dxds

(6.3)

R2

0

≤ Hint (0; R) + C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). Once Proposition 6.1 is shown, since the second term on the left hand side of (6.3) is nonnegative, we see that for all 0 < t < T and all R ≥ 28 R0 , Hint (t; R) ≤ Hint (0; R) + C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R).

(6.4)

We now prove Proposition 6.1. Firstly, we begin with the following lemma. Lemma 6.2.

It holds that d Hint (t; R) + dt

∫

2

u(t)|∇(log u(t) − ψ(t))| φR dx = R2

3 ∑

Fi (t),

(6.5)

i=1

where ∫ u(t) log u(t)(∆φR + ⟨∇ψ(t), ∇φR ⟩) dx, ∫ F2 (t) = − u(t)ψ(t)⟨∇ψ(t), ∇φR ⟩ dx + (1 + ψ(t))⟨∇u(t), ∇φR ⟩ dx,

F1 (t) =

2 R∫

R2

F3 (t) =

2 ∑ j=1

(−1)j βj

R2

{

1 d 4 dt

∫ R2

vj2 (t)∆φR dx +

} ∂t vj (t)⟨∇vj (t), ∇φR ⟩ dx .

∫ R2

Proof . The proof is done by similar calculations to those in [13, Lemma 3.13]. Using ∂t u = ∆u − ∇ · (u∇ψ), where ψ = β1 v1 − β2 v2 , and integrating by parts, we have that ∫ ∫ ∫ d 2 (u log u − u)φR dx + u|∇ log u| φR dx − ⟨∇u, ∇ψ⟩φR dx dt R2 R2 R2 ∫ ∫ ∫ = u log u∆φR dx + ⟨∇u, ∇φR ⟩ dx + u log u⟨∇ψ, ∇φR ⟩ dx. R2

R2

R2

Using ∂t u = ∆u − ∇ · (u∇ψ) again and integrating by parts yield that for j = 1, 2, ∫ ∫ ∫ ∂t uvj φR dx = − ⟨∇u, ∇vj ⟩φR dx − vj ⟨∇u, ∇φR ⟩ dx 2 R2 R∫ R2 ∫ + u⟨∇ψ, ∇vj ⟩φR dx + uvj ⟨∇ψ, ∇φR ⟩ dx. R2

(6.6)

R2

(6.7)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Next, by −∆vj = u − λj vj (j = 1, 2), ∫ ∫ ∫ 1 d λj d 2 u∂t vj φR dx = |∇vj | φR dx + v 2 φR dx 2 dt R2 2 dt R2 j R2 ∫ + ∂t vj ⟨∇vj , ∇φR ⟩ dx.

19

(6.8)

R2

Integrating by parts and using −∆vj + λj vj = u, we observe that ∫ ∫ ∫ ∫ 1 2 |∇vj | φR dx + λj vj2 ∆φR dx, vj2 φR dx = uvj φR dx + 2 2 2 2 2 R R R R and substituting this relation into the first two terms on the right-hand side of (6.8), we obtain that ∫ ∫ ∫ 1 d 1 d u∂t vj φR dx = uvj φR dx + v 2 ∆φR dx 2 dt R2 4 dt R2 j R2 ∫ (6.9) + ∂t vj ⟨∇vj , ∇φR ⟩ dx. R2

Adding (6.7) and (6.9) yields that ∫ ∫ ∫ 1 d uvj φR dx + ⟨∇u, ∇vj ⟩φR dx − u⟨∇ψ, ∇vj ⟩φR dx 2 dt R2 R2 R2 ∫ ∫ =− vj ⟨∇u, ∇φR ⟩ dx + uvj ⟨∇ψ, ∇φR ⟩ dx 2 R2 ∫ ∫ R 1 d + v 2 ∆φR dx + ∂t vj ⟨∇vj , ∇φR ⟩ dx. 4 dt R2 j R2 As ψ = β1 v1 − β2 v2 , it follows from (6.10) that ∫ ∫ ∫ 1 d 2 uψφR dx + ⟨∇u, ∇ψ⟩φR dx − u|∇ψ| φR dx 2 dt R2 2 2 R R ∫ ∫ =− ψ⟨∇u, ∇φR ⟩ dx + uψ⟨∇ψ, ∇φR ⟩ dx R2

(6.10)

(6.11)

R2

∫ 2 {1 d ∫ } ∑ j 2 (−1) βj − vj ∆φR dx + ∂t vj ⟨∇vj , ∇φR ⟩ dx . 4 dt R2 R2 j=1 Subtracting (6.11) from (6.6) yields that ∫ ∫ } d{ 1 (u log u − u)φR dx − uψφR dx dt R2 2 R2 ∫ ∫ ∫ 2 2 + u|∇ log u| φR dx − 2 ⟨∇u, ∇ψ⟩φR dx + u|∇ψ| φR dx 2 2 2 R R R ∫ ∫ ∫ = u log u∆φR dx + ⟨∇u, ∇φR ⟩ dx + u log u⟨∇ψ, ∇φR ⟩ dx 2 R∫ R2 ∫ R2 + ψ⟨∇u, ∇φR ⟩ dx − uψ⟨∇ψ, ∇φR ⟩ dx R2 2 ∑

R2

∫ } {1 d ∫ + (−1)j βj vj2 ∆φR dx + ∂t vj ⟨∇vj , ∇φR ⟩ dx . 4 dt R2 R2 j=1 As ⟨∇u, ∇ψ⟩ = u⟨∇ log u, ∇ψ⟩ on the left-hand side of (6.12), we have 2

2

2

u|∇ log u| φR − 2⟨∇u, ∇ψ⟩φR + u|∇ψ| φR = u|∇(log u − ψ)| φR , and hence, we establish (6.5). □

(6.12)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

20

The following two lemmas are to give the estimates for Fi (t) (i = 1, 2, 3) appearing on the right hand side of (6.5) in Lemma 6.2. Lemma 6.3.

Let R ≥ 25 R0 . Then ∫ T (|F1 (t)| + |F2 (t)|) dt ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T, R).

(6.13)

0

Proof . We show only the estimate for F1 (t) since F2 (t) is estimated in a similar way. Firstly we note that supp ∇φR , supp ∆φR ⊂ {x ∈ R2 | R ≤ |x| ≤ 2R}. Using { 1/e, 0 < u ≤ 1, 0 ≤ u| log u| ≤ u2 , u ≥ 1 and |∆φR | ≤ CR−2 by virtue of (6.1), we have that for all 0 < t < T , ∥u(t) log u(t)∆φR ∥1 ∫ ∫ = u(t)| log u(t)∥∆φR | dx + u(t)| log u(t)∥∆φR | dx 0
0 < t < T.

Hence, we conclude from (5.11) and (5.19) that ∫ T |F1 (s)| ds ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T, R). □ 0

Lemma 6.4.

For all R ≥ 28 R0 , the following estimate holds: ⏐∫ t ⏐ ⏐ ⏐ ⏐ sup ⏐ F3 (s) ds⏐⏐ ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). 0
0

Proof . Let JR ∈ C0∞ (R2 ) satisfy the following properties: ⎧ ⎪ R ≤ |x| ≤ 2R, ⎨= 1, JR (x) ∈ (0, 1), R/2 < |x| < R, 2R < |x| < 3R, ⎪ ⎩ 0, otherwise, |∇JR | ≤ CR−1 ,

|∆JR | ≤ CR−2 ,

and then put φˆR = JR φR . Since φR = φˆR (R ≤ |x| ≤ 2R), we can represent F3 (t) as follows: F3 (t) =

2 ∑ (−1)j βj F3,j (t), j=1

where F3,j (t) =

1 d 4 dt

∫ R2

vj2 (t)∆φˆR dx +

∫ ∂t vj (t)⟨∇vj (t), ∇φˆR ⟩ dx. R2

(6.14)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

21

By calculations similar to those in [13, Lemma 3.15], we see that 1 2 F3,j (t) =: F3,j (t) + F3,j (t),

where ∫

1 F3,j (t)

∫

2

=−β vj (t)u (t)φˆR dx + (λ1 β1 v1 (t) − λ2 β2 v2 (t))vj (t)u(t)φˆR dx R2 ∫ ∫ R2 + ⟨∇u(t), ∇vj (t)⟩φˆR dx + vj (t)⟨∇u(t), ∇ψ(t)⟩φˆR dx 2 R2 ∫R + vj (t)⟨∇u(t), ∇φˆR ⟩ dx (β = β1 − β2 ), 2 R (∫ ) ∫ ∫ 1 d 1 2 2 F3,j (t) = |∇vj (t)| φˆR dx + λj vj2 (t)∆φˆR dx . vj2 (t)φˆR dx − 2 dt 2 R2 R2 R2 Here we have used 2∂t vj ⟨∇vj , ∇φˆR ⟩ =2∇ · (∂t vj vj ∇φˆR − vj ∇∂t vj φˆR − vj ∇uφˆR ) − 2βvj u2 φˆR + 2(λ1 β1 v1 − λ2 β2 v2 )vj uφˆR + 2⟨∇u, ∇vj ⟩φˆR + 2vj ⟨∇u, ∇ψ⟩φˆR + 2vj ⟨∇u, ∇φˆR ⟩ 2

+ ∂t (|∇vj | φˆR + λj vj2 φˆR − vj2 ∆φˆR ). Since supp φˆR ⊂ {x ∈ R2 | R/2 ≤ |x| ≤ 2R}, using the estimate (5.2) on the L∞ -norm of vj , (5.10) and (5.11) on the Lp -norms (3/2 ≤ p ≤ ∞) of ∇vj and ∇ψ and (5.19) and (5.28) on L∞ -norms of u and ∇u, we obtain that ⏐∫ t ⏐ ⏐ ⏐ 1 F3,j (s) ds⏐ ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R), 0 < t < T. ⏐ 0

Also, since vj (0), ∇vj (0) ∈ L2 (R2 ) due to vj (0) = (λj − ∆)−1 u0 (j = 1, 2) and u0 ∈ L1 ∩ L∞ by applying Lemma 2.2, by (3.3) and (5.10), we have that for all R ≥ 28 R0 and all 0 < t < T , ⏐∫ t ⏐ ⏐ ⏐ 2 ⏐ F3,j (s) ds⏐⏐ ≤ C + C∥vj (t)∥2L2 (|x|≥R/2) + C∥∇vj (t)∥2L2 (|x|≥R/2) ⏐ 0

≤ C(∥u0 ∥1 , ∥u0 ∥3 , T ). Consequently, combining these estimates above, we obtain that for all R ≥ 28 R0 , ⏐∫ t ⏐ ⏐ ⏐ sup ⏐ F3,j (s) ds⏐ ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). 0
0

Thus, (6.14) is established. □ Proof of Proposition 6.1. Integrating (6.5) from 0 to t with respect to the time variable, we obtain that ∫ t∫ 3 ∫ t ∑ 2 Hint (t; R) + u|∇(log u − ψ)| φR dxds = Hint (0; R) + Fj (s) ds. 0

R2

j=1

0

Therefore, by Lemmas 6.3 and 6.4, we obtain the desired estimate (6.3). □ 7. A priori estimates for entropy in interior domains In this section we are going to give an a priori estimate for the modified entropy in the interior domain. For this purpose, we introduce F(t; R) as ∫ ∫ 1 u(t)ψ(t)φR dx, ψ = β1 v1 − β2 v2 , (7.1) F(t; R) := u(t) log u(t)φR dx − 2 R2 R2 where the function φR is the one defined by (6.1). First of all, we begin with the following lemma.

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

22

Lemma 7.1. Let R ≥ 28 R0 . Then sup F(t; R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R).

(7.2)

0
Proof . Since 0 ≤ φR ≤ 1 and u log u − u ≤ u log u ≤ u2 /e (u > 0) by log u/u ≤ 1/e (u > 0), we have that ∫ (u0 log u0 − u0 )φR dx ≤ C∥u0 ∥22 . R2

Also, observing that ∥vj (0)∥p = ∥(λj − ∆)−1 u0 ∥p ≤ C∥u0 ∥1 (1 ≤ p < ∞) by the application of (2.2), we ∫ have that 1 − u(0)ψ(0)φR dx ≤ C∥u0 ∥2 ∥β1 v1 (0) − β2 v2 (0)∥2 ≤ C∥u0 ∥1 ∥u0 ∥2 . 2 R2 Hence, Hint (0; R) ≤ C(∥u0 ∥1 , ∥u0 ∥2 ), from which together with (6.4) it follows that sup Hint (t; R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). 0
By the definitions of Hint (t; R) and F(t; R) and the estimate above, ∫ F(t; R) = Hint (t; R) + u(t)φR dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). R2

Here we used Lemma 7.2.

∫

R2

u(t)φR dx ≤ ∥u0 ∥1 . Consequently we can get the desired estimate (7.2). □

Let R ≥ 28 R0 . Then sup F1 (t; R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R),

(7.3)

0
where β = β1 − β2 and ∫

β F1 (t; R) := u(t) log u(t)φR dx − 2 u(t)≥1

∫ u(t)v1 (t)φR dx. u(t)≥1

Proof . Noting that −u log u ≤ 1/e (0 < u ≤ 1) and 0 ≤ φR ≤ 1, by (3.3) and (7.2), we observe that ∫ ∫ 1 u(t) log u(t)φR dx − u(t)ψ(t)φR dx 2 u(t)≥1 u(t)≥1 ∫ ∫ 1 u(t)ψ(t)φR dx (7.4) = F(t; R) + (−u(t) log u(t))φR dx + 2 0
T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

Hence, 1 2

∫ u(t)ψ(t)φR dx ≤ u(t)≥1

β 2

∫

23

u(t)v1 (t)φR dx + C∥u0 ∥21 .

u(t)≥1

Substituting this inequality into the second integral term on the left-hand side of (7.4), we conclude (7.3). □ Lemma 7.3. Let R be a constant satisfying R ≥ 28 R0 and let 0 < θ < 1. Then, for all 0 < t < T , it holds that ( ) ∫ ∫ 1−θ βv1 (t) 1 exp dx. (7.5) u(t) log u(t)φR dx ≤ F1 (t; R) + θ θ 2(1 − θ) |x|<2R u(t)≥1 Proof . Using the elementary inequality uv ≤ u log u−u+ev (u ≥ 1, v ≥ 0), for u ≥ 1, v ≥ 0 and 0 < θ < 1, we observe that { ( )} βv1 β βv1 uv1 = (1 − θ)u · ≤ (1 − θ) u log u − u + exp 2 2(1 − θ) 2(1 − θ) ( ) βv1 ≤ u log u − θu log u + (1 − θ) exp , 2(1 − θ) from which it follows that 1 u log u ≤ θ

( ( ) ) βv1 β 1−θ u log u − uv1 + exp . 2 θ 2(1 − θ)

(7.6)

Therefore, multiplying (7.6) by φR and then integrating it over {x ∈ R2 | u(t) ≥ 1}, we obtain ( ) ∫ ∫ 1 1−θ βv1 (t) u(t) log u(t)φR dx ≤ F1 (t; R) + exp φR dx θ θ 2(1 − θ) u(t)≥1 u(t)≥1 ( ) ∫ 1 1−θ βv1 (t) ≤ F1 (t; R) + exp dx. θ θ 2(1 − θ) |x|<2R Thus, (7.5) is established. □ By Lemma 7.2, the first term on the right-hand side of (7.5) is estimated as follows: 1 1 F1 (t; R) ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). θ θ

(7.7)

For the second term, by applying the Brezis–Merle type inequality (Corollary 2.7), the following estimate holds: Lemma 7.4. Let R be a constant satisfying R ≥ 28 R0 . Then, there exists a constant θ with 0 < θ < 1 such that for all 0 < t < T , ( ) ∫ βv1 (t) dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , T, R, θ). (7.8) exp 2(1 − θ) |x|<2R To show Lemma 7.4, we need the following estimate of u from below stated in Lemma 7.5. This lemma is proven by applying the strong maximum principle for parabolic equations, thanks to the boundedness of ∂xα u (|α| ≤ 2) on [0, T ) × {|x| ≥ R} stated in Propositions 5.3 and 5.5. We omit its proof, because the proof of the lemma is almost the same as that of [13, Proposition 3.12]. Lemma 7.5. Let R be a constant with R ≥ 28 R0 . Then, there exist x0 ∈ R2 satisfying |x0 | ≥ 3R, ε0 ∈ (0, 1] and d0 > 0 such that u(t, x) ≥ d0 (0 ≤ t < T, |x − x0 | ≤ ε0 ). (7.9)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

24

Proof of Lemma 7.4. Let x0 , ε0 and d0 be the ones as in Lemma 7.5 for which the following is satisfied: u(t, x) ≥ d0

(0 ≤ t < T, |x − x0 | ≤ ε0 ).

For 0 < θ < 1, put w := βv1 /{2(1 − θ)} ≥ 0 and g := βu/{2(1 − θ)}. The function w satisfies −∆w + λ1 w = g (|x| < 2R) due to −∆v1 + λ1 v1 = u. Remarking that {|x| ≤ 2R} ∩ {|x − x0 | ≤ ε0 } = ϕ, we have ∫ ∫ ∫ u(t) dx = u(t) dx − u(t) dx |x|<2R R2 |x|≥2R ) ( ∫ ∫ ∫ 8 8π − d0 − d0 ε20 π. dx = ≤ u0 dx − u(t) dx ≤ β β |x−x0 |≤ε0 R2 |x−x0 |≤ε0 Hence,

∫ g(t) dx = |x|<2R

β 2(1 − θ)

∫ u(t) dx ≤ |x|<2R

π (8 − βd0 ε20 ), 2(1 − θ)

from which it follows that ∫

π (8 − βd0 ε20 ) 2(1 − θ) |x|<2R { } π π = 8(1 − θ) − (8 − βd0 ε20 ) = (βd0 ε20 − 8θ). 2(1 − θ) 2(1 − θ)

4π −

g(t) dx ≥ 4π −

Taking θ ∈ (0, 1) such that βd0 ε20 − 8θ > 0, we have that ∫ g(t) dx < 4π. |x|<2R

Therefore, Corollary 2.7 ensures that for all 0 < t < T , ( ) ∫ ∫ βv1 (t, x) exp dx = ew(t,x) dx 2(1 − θ) |x|<2R |x|<2R { } 2 4π 2 ≤ (4R) exp sup w(t, x) 4π − ∥g(t)∥L1 (|x|<2R) |x|=2R ) ( 128πR2 (1 − θ) β sup v1 (t, x) ≤ exp βd0 ε20 − 8θ 2(1 − θ) |x|=2R ( ) β 128πR2 (1 − θ) exp sup ∥v1 (t)∥L∞ (|x|≥2R) . ≤ βd0 ε20 − 8θ 2(1 − θ) 0
Let R ≥ 28 R0 . Then ∫ sup (1 + u(t)) log(1 + u(t)) dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R).

0
(7.10)

|x|≤R

Proof . Fix θ ∈ (0, 1) for which (7.8) is satisfied. We claim that ∫ u(t)| log u(t)| dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R),

0 < t < T.

|x|≤R

Indeed, since φR (x) = 1 (|x| ≤ R), we see that ∫ ∫ u(t)| log u(t)| dx ≤ u(t)| log u(t)|φR dx |x|≤R R2 ∫ ∫ = u(t) log u(t)φR dx + {u(t)≥1}∩{|x|≤2R}

{0
(−u(t) log u(t))φR dx.

(7.11)

T. Nagai and T. Yamada / Nonlinear Analysis 190 (2020) 111615

25

As the right-hand side of (7.5) is estimated by a positive constant depending on ∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R thanks to (7.7) and (7.8), we have that for all 0 < t < T , ∫ u(t) log u(t)φR dx {u(t)≥1}∩{|x|≤2R} ∫ ≤ u(t) log u(t)φR dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). u(t)≥1

Next, by −u log u ≤ 1/e (0 < u ≤ 1) and 0 ≤ φR (x) ≤ 1 (x ∈ R2 ), we have that for all 0 < t < T , ∫ ∫ 4πR2 1 dx ≤ . (−u(t) log u(t))φR dx ≤ e {0
□

(1 + u(t)) log(1 + u(t)) dx.

|x|≥28 R0

From (3.5) and (7.10), we have ∫ sup (1 + u(t)) log(1 + u(t)) dx ≤ C(∥u0 ∥1 , ∥u0 ∥∞ , ∥∇u0 ∥2 , T, R). 0
R2

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