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International Journal of Differential Equations
Volume 2011 (2011), Article ID 896427, 9 pages
http://dx.doi.org/10.1155/2011/896427
Research Article

A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System

1School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3School of Mathematics & Computational Science, Sun Yat-sen University, Guangzhou 510275, China

Received 30 May 2011; Accepted 7 September 2011

Academic Editor: Sining Zheng

Copyright © 2011 Guocai Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable on . As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

1. Introduction

In this paper, we are concerned with the following two problems: one is blow-up rates for blow-up solutions of the higher-order semilinear parabolic system 𝑢𝑡+(Δ)𝑚𝑢=|𝑣|𝑝,𝑣𝑡+(Δ)𝑚𝑣=|𝑢|𝑞,(𝑥,𝑡)𝑁×(0,𝑇),𝑢(𝑥,0)=𝑢0(𝑥)𝐿𝑁,𝑣(𝑥,0)=𝑣0(𝑥)𝐿𝑁,(1.1) where 𝑚1 and 𝑝,𝑞>1; the other is parabolic Liouville theorems for the problem 𝑢𝑡+(Δ)𝑚𝑢=|𝑣|𝑝,𝑣𝑡+(Δ)𝑚𝑣=|𝑢|𝑞,(𝑥,𝑡)𝑁×,𝑢(𝑥,𝑡)𝐿𝑝loc𝑁+1,𝑣(𝑥,𝑡)𝐿𝑞loc𝑁+1,(1.2) where 𝑚1 and 𝑝,𝑞>1. The first problem is directly related to the second one. Actually, blow-up rates of the blow-up solutions, by scaling arguments, are often converted to nonexistence of solutions of some limiting problems with 𝑡 (see, e.g., Poláčik and Quittner [1] and Xing [2]).

Recall that, in his famous paper [3], Fujita studied the initial value problem 𝑢𝑡Δ𝑢=𝑢𝑝,(𝑥,𝑡)𝑁×(0,),𝑢(𝑥,0)=𝑢0(𝑥),𝑥𝑁,(1.3) for nonnegative initial data 𝑢0. He obtained the following.(i)If 1<𝑝<1+2/𝑁, then the only nonnegative global solution is 𝑢0.(ii)If 𝑝>1+2/𝑁, then there exist global solutions for some small initial value.

The number 1+2/𝑁 belonging to Case (i) had been answered in [47], and an elegant proof was given by Weissler [7]. The number 1+2/𝑁 is named the critical blow-up exponent (or critical Fujita exponent).

Ever since then, Fujita's result has been given great attention and extended in various directions. One direction is to consider the problems on other domains. For example, 𝑁 is replaced by a cone or exterior of a bounded domain, and so forth. Another direction is to extend these results to more general equations and systems (see the survey papers [810] and references therein). We just briefly describe the results directly connected to our problems.(1)The systems from the point of view of the critical blow-up exponent originate in the 1990s. Escobedo and Herrero [11] discussed the following weakly coupled second-order parabolic system 𝑢𝑡Δ𝑢=𝑣𝑝,𝑣𝑡Δ𝑣=𝑢𝑞,(𝑥,𝑡)𝑁×(0,𝑇),𝑢(𝑥,0)=𝑢0(𝑥)0𝐿𝑁,𝑣(𝑥,0)=𝑣0(𝑥)0𝐿𝑁,(1.4) with 𝑝>0 and 𝑞>0. They established the following.(i)If 0<𝑝𝑞1, then all solutions are global.(ii)If 𝑝𝑞>1 and 𝑁/2max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then every nontrivial solution blows up in finite time.(iii)If 𝑝𝑞>1 and 𝑁/2>max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then there exist both global solutions and blow-up solutions.(2)Egorov et al. [12] considered a class of higher-order parabolic system of inequalities and gave some results about nonexistence of the nontrivial global solutions with initial data having nonnegative average value.(3)A natural generalization of classical weakly coupled system (1.4) are the higher-order parabolic system (1.1). Pang et al. [13] studied (1.1) and obtained the following results.(i)If 𝑁/2𝑚min{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then every solution with initial data having positive average value does not exist globally in time.(ii)If 𝑁/2𝑚>max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then global solutions with small initial data exist. Notice that there exists a gap between the range of exponent in the two cases. In fact, in an earlier monograph [14], Mitidieri and Pokhozhaev have shown that Case (i) holds true for 𝑁/2𝑚max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)} (see [14, Example  38.2]). Integrating these results in [13, 14], one directly obtains a complete Fujita-type theorem for the higher-order parabolic system (1.1).

Theorem 1.1. Assume 𝑝>1 and 𝑞>1. Then(i)if 𝑁/2𝑚max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then every solution of (1.1) with initial data having positive average value does not exist globally in time;(ii)if 𝑁/2𝑚>max{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}, then global solutions of (1.1) with small initial data exist.

(4)Recently, Pan and Xing [15] considered the problem (1.2) and proved a parabolic Liouville theorem; that is, if 𝑁/2𝑚min{(𝑞+1)/(𝑝𝑞1),(𝑝+1)/(𝑝𝑞1)}, then the global solution of (1.2) is trivial. As an immediate application of the result, blow-up rates for the problem (1.1) is also obtained: Let (𝑢,𝑣) be a solution of (1.1) which blows up at a finite time 𝑇. Then there is a constant 𝐶>0 such that sup𝑥𝑁|𝑢(𝑥,𝑡)|,sup𝑥𝑁|𝑣(𝑥,𝑡)|𝐶(𝑇𝑡)(𝑞+1)/(𝑝𝑞1) for 𝑁/2𝑚min{(𝑝+1)/(𝑝𝑞1),(𝑞+1)/(𝑝𝑞1)}.

The purpose of this note is to improve the results of [15]. More precisely, we will extend the exponent range from 𝑁/2𝑚min{(𝑞+1)/(𝑝𝑞1),(𝑝+1)/(𝑝𝑞1)} to 𝑁/2𝑚max{(𝑞+1)/(𝑝𝑞1),(𝑝+1)/(𝑝𝑞1)} for both blow-up rates and parabolic Liouville theorems of [15]. The main results of this paper are Theorems 2.1 and 3.1. Our methods are similar to [14, 15]. In fact, the present Theorem 2.1 will be proved by modifying part of the proof of Theorem  4.3 of [15].

The organization of this paper is as follows. In Section 2, we improve the range of the exponent for parabolic Liouville-type theorems in [15]. As a direct application, the exponent range in blow-up rates for corresponding systems is extended in Section 3.

2. Parabolic Liouville-Type Theorem for Higher-Order System of Inequalities

In this section, we will improve the exponent range of some parabolic Liouville-type theorems for higher-order semilinear parabolic systems.

Now we consider a class of more general parabolic systems of inequalities than (1.2). Let 𝐿=𝐿(𝑡,𝑥,𝐷𝑥) be a differential operator of order : 𝐿[𝑣]=|𝛼|=𝐷𝛼𝑎𝛼(𝑡,𝑥,𝑣)𝑣,(2.1) and let 𝑀 be a differential operator of order : 𝑀[𝑣]=||𝛽||=𝐷𝛽𝑏𝛽(𝑡,𝑥,𝑣)𝑣,(2.2) where 𝑎𝛼(𝑡,𝑥,𝑣) and 𝑏𝛽(𝑡,𝑥,𝑣) are bounded functions defined for 𝑡,𝑥𝑁,and𝑣.

Consider the set of (𝑢,𝑣) satisfying the inequalities: 𝜕𝑢[𝑢]𝜕𝑡𝐿+|𝑣|𝑞2,𝜕𝑣[𝑣]𝜕𝑡𝑀+|𝑢|𝑞1,(𝑥,𝑡)𝑁×,(2.3)

in the following weak sense: if 𝜓𝐶0max{,}(𝑁+1) and 𝜓(𝑥,𝑡)0, then 𝜕𝜓𝜕𝑡𝑢𝑑𝑥𝑑𝑡𝑢𝐿[𝜓]|𝑑𝑥𝑑𝑡𝑣|𝑞2𝜓𝑑𝑥𝑑𝑡,(2.4)𝜕𝜓𝜕𝑡𝑣𝑑𝑥𝑑𝑡𝑣𝑀[𝜓]𝑑𝑥𝑑𝑡|𝑢|𝑞1𝜓𝑑𝑥𝑑𝑡.(2.5) Here and in the following, if the limits of integration are not given, then the integrals are taken over the space 𝑁×, and 𝐿[𝜓]=|𝛼|=𝑎𝛼(𝑡,𝑥,𝑢)(𝐷)𝛼𝜓,𝑀[𝜓]=|𝛼|=𝑏𝛼(𝑡,𝑥,𝑢)(𝐷)𝛼𝜓.(2.6)

Here is the main result of this section.

Theorem 2.1. If two functions 𝑢(𝑥,𝑡)𝐿𝑞1loc(𝑁+1) and 𝑣(𝑥,𝑡)𝐿𝑞2loc(𝑁+1) satisfy (2.4) and (2.5), then 𝑢(𝑥,𝑡)0,𝑣(𝑥,𝑡)0 for 𝑞1,𝑞2>1 and (𝑞1,𝑞2)Γ1Γ2, where Γ1=𝑞1,𝑞2𝑁𝑞min{,}max1+1𝑞1𝑞2,𝑞12+1𝑞1𝑞2,Γ12=𝑞1,𝑞2𝑞𝑁+max{,}max+1+𝑞1𝑞2𝑞1,+2+𝑞1𝑞2.1(2.7)

Remark 2.2. In fact, we will extend the range of the exponents 𝑞1,𝑞2 in Theorem 4.3 of [15] from 𝑞1,𝑞2𝑁𝑞min{,}min1+1𝑞1𝑞2,𝑞12+1𝑞1𝑞21(2.8) to Γ1Γ2. Obviously, Γ1 contains the range of the exponent in [15].

As an immediate application, we take 𝐿=𝑀=(Δ)𝑚,𝑔1(𝑢,𝑣)=|𝑣|𝑝,and𝑔2(𝑢,𝑣)=|𝑢|𝑞.

Corollary 2.3. If two functions 𝑢(𝑥,𝑡)𝐿𝑝loc(𝑁+1) and 𝑣(𝑥,𝑡)𝐿qloc(𝑁+1) satisfy 𝑢𝑡+(Δ)𝑚𝑢=|𝑣|𝑝,𝑣𝑡+(Δ)𝑚𝑣=|𝑢|𝑞,(2.9) on 𝑁×, then 𝑢(𝑥,𝑡)0,𝑣(𝑥,𝑡)0 for 𝑝,𝑞>1 belonging to the following set: 𝑁(𝑝,𝑞)2𝑚max𝑞+1,𝑝𝑞1𝑝+1𝑝𝑞1.(2.10)

Remark 2.4. In fact, the present Theorem 2.1 will be proved by modifying part of the proof of Theorem  4.3 of [15]. In the following proof, the part before the inequalities (2.24) is the same as that in Theorem 4.3 of [15]. The main difference between the proofs is the discussion of the four cases in the last part of the proof. For completeness of arguments as well as convenience of readers, we give a detailed proof of the theorem.

Proof of Theorem 2.1.. Let 𝜙𝐶0max{,}(),𝜙0, and 1𝜙(𝑠)=as𝑠1,0as𝑠2.(2.11) Suppose that there exists a positive constant 𝐶 such that ||𝜙||(𝑠)𝐶𝜙1/𝑞1||𝜙(𝑠),()||(𝑠)𝐶𝜙1/𝑞1||𝜙(𝑠),||(𝑠)𝐶𝜙1/𝑞2||𝜙(𝑠),()||(𝑠)𝐶𝜙1/𝑞2(𝑠).(2.12)
In order to find such a function, one also assume that, for 3/2<𝑠<2, 𝜙(𝑠)=(2𝑠)𝛿 with 𝛿>max{𝑞1/(𝑞11),𝑞2/(𝑞21)}.
Let 𝜓𝑅(𝑥,𝑡)=𝜙|𝑡|2+|𝑥|2𝜎𝑅2𝜎,𝑅>0,(2.13) the value of the parameter 𝜎>0 will be determined below. Now putting 𝜓=𝜓𝑅(𝑥,𝑡) in (2.4) and (2.5) and letting II=|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡,III=|𝑣|𝑞2𝜓𝑅𝑑𝑥𝑑𝑡,(2.14) we have III𝜕𝜓𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡𝑢𝐿𝜓𝑅𝑑𝑥𝑑𝑡,(2.15)II𝜕𝜓𝑅𝜕𝑡𝑣𝑑𝑥𝑑𝑡𝑣𝑀𝜓𝑅𝑑𝑥𝑑𝑡.(2.16)
The Hölder inequality implies 𝜕𝜓𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡𝐶0supp𝜕𝜓𝑅𝜕𝑡|𝑢|𝜓1/𝑞1𝑅|𝑡|𝑅2𝜎𝑑𝑥𝑑𝑡𝐶1supp𝜕𝜓𝑅𝜕𝑡|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1supp𝜕𝜓𝑅1𝜕𝑡𝑅𝜎𝑞1𝑑𝑥𝑑𝑡1/𝑞1𝐶2supp𝜕𝜓𝑅𝜕𝑡|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑅𝑁+𝜎𝜎𝑞1/(𝑞11)(𝑞11)/𝑞1,𝑢𝐿𝜓𝑅𝑑𝑥𝑑𝑡𝐶3Υ1|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑅𝑁+𝜎𝑞1/(𝑞11)(𝑞11)/𝑞1,(2.17) where Υ1={(𝑥,𝑡)𝑡,𝐷𝛼𝑥𝜓𝑅(𝑥,𝑡)0forsome𝛼} and 𝑞1=𝑞1/(𝑞11). It is essential here that the operator 𝐿 contains the derivatives of order only. It is obvious that supp𝜕𝜓𝑅𝜕𝑡Υ1Σ{(𝑥,𝑡)𝑡,|𝑡|2+|𝑥|2𝜎>𝑅2𝜎}, and therefore inequality (2.15) implies that III𝐶4Σ|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑅(𝑁+𝜎)(𝑞11)/𝑞1𝑅𝜎+𝑅(2.18)𝐶4Σ|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑅𝐴(2.19)𝐶4II1/𝑞1𝑅𝐴(2.20) with 𝐴=(𝑁+𝜎)(𝑞11)/𝑞1min{𝜎,}. Similarly, II𝐶5Σ|𝑣|𝑞2𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞2𝑅(𝑁+𝜎)(𝑞21)/𝑞2𝑅𝜎+𝑅(2.21)𝐶5Σ|𝑣|𝑞2𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞2𝑅𝐵(2.22)𝐶5III1/𝑞2𝑅𝐵(2.23) with 𝐵=(𝑁+𝜎)(𝑞21)/𝑞2min{𝜎,}. Then (2.20) and (2.23) lead to II(𝑞1𝑞21)/𝑞1𝑞2𝐶6𝑅𝐵+𝐴1/𝑞2,III(𝑞1𝑞21)/𝑞1𝑞2𝐶7𝑅𝐴+𝐵1/𝑞1.(2.24)
We consider the following cases. Case 1. 𝐵+𝐴(1/𝑞2)<0. Let 𝑅+ in the first inequality of (2.24), we obtain |𝑢|𝑞1𝑑𝑥𝑑𝑡=0,(2.25) which implies 𝑢0. Combining with inequality (2.20) or equality (2.4), we get that |𝑣|𝑞2𝑑𝑥𝑑𝑡=0. Then 𝑣0.Case 2. 𝐴+𝐵(1/𝑞1)<0. Inequality (2.24) implies 𝑣0. The inequality (2.23) or equality (2.5) leads that 𝑢0.Case 3. 𝐵+𝐴(1/𝑞2)=0. By (2.22) and (2.19), we have II𝐶8Σ|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑞2𝑅𝐵+𝐴1/𝑞2=𝐶8Σ|𝑢|𝑞1𝜓𝑅𝑑𝑥𝑑𝑡1/𝑞1𝑞2.(2.26) And, from (2.24), we obtain that |𝑢|𝑞1𝑑𝑥𝑑𝑡 converge. Then, as 𝑅+, Σ|𝑢|𝑞1𝑑𝑥𝑑𝑡0.(2.27) Then 𝑢0 and (2.20) implies 𝑣0.Case 4. 𝐴+𝐵(1/𝑞1)=0. Similarly to Case 3, the second inequality of (2.24) implies |𝑣|𝑞𝑑𝑥𝑑𝑡 converging. 𝑣0 follows from (2.19) and (2.22). Then by (2.23) or (2.5), 𝑢0.
Taking 𝜎=min{,}, Cases 1 and 3: 𝐵+𝐴(1/𝑞2)0 is equivalent to 𝑁/min{,}(𝑞1+1)/(𝑞1𝑞21), Cases 2 and 4: 𝐴+𝐵(1/𝑞1)0 is 𝑁/min{,}(𝑞2+1)/(𝑞1𝑞21). So the union of Cases 14 is just the set Γ1.
Similarly, taking 𝜎=max{,}, we obtain that the union of Cases 14 is equivalent to the set Γ2. Then we get the result.

3. Blow-Up Rate Estimates for Parabolic Systems

As an immediate application of Corollary 2.3, the range of the exponents 𝑝,𝑞 in blow-up rates for the system (1.1) is also extended. We have the following theorem.

Theorem 3.1. Let (𝑢,𝑣) be a solution of (1.1) which blows up at a finite time 𝑇. Then there is a constant 𝐶>0 such that sup𝑥𝑁||||𝑢(𝑥,𝑡)𝐶(𝑇𝑡)(𝑝+1)/(𝑝𝑞1),sup𝑥𝑁||||𝑣(𝑥,𝑡)𝐶(𝑇𝑡)(𝑞+1)/(𝑝𝑞1)(3.1) for 𝑁2𝑚max𝑝+1,𝑝𝑞1𝑞+1𝑝𝑞1.(3.2)

Since the proof of Theorem 3.1 is completely similar to Theorem 3.1 in [15], we omit it. Refer to [15] for all the details.

Acknowledgment

The first author was supported by Fundamental Research Funds for the Central Universities (Grant no. 2010121006). The second author was supported by NSFC (Grant no. 10901059). The third author was supported by NSFC (Grant nos. 10821067 and 11001277), RFDP (Grant no. 200805581023), Research Fund for the Doctoral Program of Guangdong Province of China (Grant no. 9451027501002416), and Fundamental Research Funds for the Central Universities.

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