International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 896427 | 9 pages | https://doi.org/10.1155/2011/896427

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

Academic Editor: Sining Zheng
Received30 May 2011
Accepted07 Sep 2011
Published02 Nov 2011

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 ℝ. 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|ğ‘ž,(𝑥,𝑡)∈ℝ𝑁×(0,𝑇),𝑢(𝑥,0)=𝑢0(𝑥)âˆˆğ¿âˆžî€·â„ğ‘î€¸,𝑣(𝑥,0)=𝑣0(𝑥)âˆˆğ¿âˆžî€·â„ğ‘î€¸,(1.1) where 𝑚⩾1 and 𝑝,ğ‘ž>1; the other is parabolic Liouville theorems for the problem 𝑢𝑡+(−Δ)𝑚𝑢=|𝑣|𝑝,𝑣𝑡+(−Δ)𝑚𝑣=|𝑢|ğ‘ž,(𝑥,𝑡)∈ℝ𝑁×ℝ,𝑢(𝑥,𝑡)∈𝐿𝑝locℝ𝑁+1,𝑣(𝑥,𝑡)âˆˆğ¿ğ‘žlocℝ𝑁+1,(1.2) where 𝑚⩾1 and 𝑝,ğ‘ž>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∈ℝ (see, e.g., Poláčik and Quittner [1] and Xing [2]).

Recall that, in his famous paper [3], Fujita studied the initial value problem 𝑢𝑡−Δ𝑢=𝑢𝑝,(𝑥,𝑡)∈ℝ𝑁×(0,∞),𝑢(𝑥,0)=𝑢0(𝑥),𝑥∈ℝ𝑁,(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 𝑢≡0.(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 [4–7], 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, ℝ𝑁 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 [8–10] 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 𝑢𝑡−Δ𝑢=𝑣𝑝,𝑣𝑡−Δ𝑣=ğ‘¢ğ‘ž,(𝑥,𝑡)∈ℝ𝑁×(0,𝑇),𝑢(𝑥,0)=𝑢0(𝑥)⩾0âˆˆğ¿âˆžî€·â„ğ‘î€¸,𝑣(𝑥,0)=𝑣0(𝑥)⩾0âˆˆğ¿âˆžî€·â„ğ‘î€¸,(1.4) with 𝑝>0 and ğ‘ž>0. They established the following.(i)If 0<ğ‘ğ‘žâ©½1, then all solutions are global.(ii)If ğ‘ğ‘ž>1 and 𝑁/2⩽max{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, then every nontrivial solution blows up in finite time.(iii)If ğ‘ğ‘ž>1 and 𝑁/2>max{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, 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/2𝑚⩽min{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, then every solution with initial data having positive average value does not exist globally in time.(ii)If 𝑁/2𝑚>max{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, 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/2𝑚⩽max{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)} (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 ğ‘ž>1. Then   (i)if 𝑁/2𝑚⩽max{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, 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)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}, 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/2𝑚⩽min{(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1),(𝑝+1)/(ğ‘ğ‘žâˆ’1)}, 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 sup𝑥∈ℝ𝑁|𝑢(𝑥,𝑡)|,sup𝑥∈ℝ𝑁|𝑣(𝑥,𝑡)|⩽𝐶(𝑇−𝑡)−(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1) for 𝑁/2𝑚⩽min{(𝑝+1)/(ğ‘ğ‘žâˆ’1),(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)}.

The purpose of this note is to improve the results of [15]. More precisely, we will extend the exponent range from 𝑁/2𝑚⩽min{(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1),(𝑝+1)/(ğ‘ğ‘žâˆ’1)} to 𝑁/2𝑚⩽max{(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1),(𝑝+1)/(ğ‘ğ‘žâˆ’1)} 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: 𝐿[𝑣]∶=|𝛼|=â„“ğ·ğ›¼î€·ğ‘Žğ›¼(𝑡,𝑥,𝑣)𝑣,(2.1) and let 𝑀 be a differential operator of order ℎ: 𝑀[𝑣]∶=||𝛽||=â„Žğ·ğ›½î€·ğ‘ğ›½(𝑡,𝑥,𝑣)𝑣,(2.2) where ğ‘Žğ›¼(𝑡,𝑥,𝑣) and 𝑏𝛽(𝑡,𝑥,𝑣) are bounded functions defined for 𝑡∈ℝ,𝑥∈ℝ𝑁,and𝑣∈ℝ.

Consider the set of (𝑢,𝑣) satisfying the inequalities: 𝜕𝑢[𝑢]𝜕𝑡⩾𝐿+|𝑣|ğ‘ž2,𝜕𝑣[𝑣]𝜕𝑡⩾𝑀+|𝑢|ğ‘ž1,(𝑥,𝑡)∈ℝ𝑁×ℝ,(2.3)

in the following weak sense: if 𝜓∈𝐶0max{ℓ,ℎ}(ℝ𝑁+1) and 𝜓(𝑥,𝑡)⩾0, then −𝜕𝜓𝜕𝑡𝑢𝑑𝑥𝑑𝑡−𝑢𝐿∗[𝜓]|𝑑𝑥𝑑𝑡⩾𝑣|ğ‘ž2−𝜓𝑑𝑥𝑑𝑡,(2.4)𝜕𝜓𝜕𝑡𝑣𝑑𝑥𝑑𝑡−𝑣𝑀∗[𝜓]𝑑𝑥𝑑𝑡⩾|𝑢|ğ‘ž1𝜓𝑑𝑥𝑑𝑡.(2.5) Here and in the following, if the limits of integration are not given, then the integrals are taken over the space ℝ𝑁×ℝ, and 𝐿∗[𝜓]∶=|𝛼|=â„“ğ‘Žğ›¼(𝑡,𝑥,𝑢)(−𝐷)𝛼𝜓,𝑀∗[𝜓]∶=|𝛼|=â„Žğ‘ğ›¼(𝑡,𝑥,𝑢)(−𝐷)𝛼𝜓.(2.6)

Here is the main result of this section.

Theorem 2.1. If two functions 𝑢(𝑥,𝑡)âˆˆğ¿ğ‘ž1loc(ℝ𝑁+1) and 𝑣(𝑥,𝑡)âˆˆğ¿ğ‘ž2loc(ℝ𝑁+1) satisfy (2.4) and (2.5), then 𝑢(𝑥,𝑡)≡0,𝑣(𝑥,𝑡)≡0 for ğ‘ž1,ğ‘ž2>1 and (ğ‘ž1,ğ‘ž2)∈Γ1∪Γ2, where Γ1=î‚»î€·ğ‘ž1,ğ‘ž2î€¸âˆ£ğ‘î‚»ğ‘žmin{ℓ,ℎ}⩽max1+1ğ‘ž1ğ‘ž2,ğ‘žâˆ’12+1ğ‘ž1ğ‘ž2,Γ−12=î‚»î€·ğ‘ž1,ğ‘ž2î€¸î‚»ğ‘žâˆ£ğ‘+max{ℓ,ℎ}⩽maxℎ+1ℓ+â„Žğ‘ž1ğ‘ž2ğ‘žâˆ’1,ℓ+2ℎ+â„“ğ‘ž1ğ‘ž2.−1(2.7)

Remark 2.2. In fact, we will extend the range of the exponents ğ‘ž1,ğ‘ž2 in Theorem 4.3 of [15] from î‚»î€·ğ‘ž1,ğ‘ž2î€¸âˆ£ğ‘î‚»ğ‘žmin{ℓ,ℎ}⩽min1+1ğ‘ž1ğ‘ž2,ğ‘žâˆ’12+1ğ‘ž1ğ‘ž2−1(2.8) to Γ1∪Γ2. 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|ğ‘ž.

Corollary 2.3. If two functions 𝑢(𝑥,𝑡)∈𝐿𝑝loc(ℝ𝑁+1) and 𝑣(𝑥,𝑡)∈𝐿qloc(ℝ𝑁+1) satisfy 𝑢𝑡+(−Δ)𝑚𝑢=|𝑣|𝑝,𝑣𝑡+(−Δ)𝑚𝑣=|𝑢|ğ‘ž,(2.9) on ℝ𝑁×ℝ, then 𝑢(𝑥,𝑡)≡0,𝑣(𝑥,𝑡)≡0 for 𝑝,ğ‘ž>1 belonging to the following set: 𝑁(𝑝,ğ‘ž)∣2𝑚⩽maxğ‘ž+1,ğ‘ğ‘žâˆ’1𝑝+1ğ‘ğ‘žâˆ’1.(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 𝜙∈𝐶0max{ℓ,ℎ}(ℝ),𝜙⩾0, and 1𝜙(𝑠)=as𝑠⩽1,0as𝑠⩾2.(2.11) Suppose that there exists a positive constant 𝐶 such that ||ğœ™î…ž||(𝑠)⩽𝐶𝜙1/ğ‘ž1||𝜙(𝑠),(ℓ)||(𝑠)⩽𝐶𝜙1/ğ‘ž1||𝜙(𝑠),||(𝑠)⩽𝐶𝜙1/ğ‘ž2||𝜙(𝑠),(ℎ)||(𝑠)⩽𝐶𝜙1/ğ‘ž2(𝑠).(2.12)
In order to find such a function, one also assume that, for 3/2<𝑠<2, 𝜙(𝑠)=(2−𝑠)𝛿 with 𝛿>max{â„“ğ‘ž1/(ğ‘ž1−1),â„Žğ‘ž2/(ğ‘ž2−1)}.
Let 𝜓𝑅(𝑥,𝑡)=𝜙|𝑡|2+|𝑥|2ğœŽğ‘…2ğœŽî‚¶,𝑅>0,(2.13) the value of the parameter ğœŽ>0 will be determined below. Now putting 𝜓=𝜓𝑅(𝑥,𝑡) in (2.4) and (2.5) and letting II=|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡,III=|𝑣|ğ‘ž2𝜓𝑅𝑑𝑥𝑑𝑡,(2.14) we have III⩽−𝜕𝜓𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡−𝑢𝐿∗𝜓𝑅𝑑𝑥𝑑𝑡,(2.15)II⩽−𝜕𝜓𝑅𝜕𝑡𝑣𝑑𝑥𝑑𝑡−𝑣𝑀∗𝜓𝑅𝑑𝑥𝑑𝑡.(2.16)
The Hölder inequality implies −𝜕𝜓𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡⩽𝐶0supp𝜕𝜓𝑅𝜕𝑡|𝑢|𝜓1/ğ‘ž1𝑅|𝑡|𝑅2ğœŽğ‘‘ğ‘¥ğ‘‘ğ‘¡â©½ğ¶1⎧⎪⎨⎪⎩supp𝜕𝜓𝑅𝜕𝑡|𝑢|ğ‘ž1ğœ“ğ‘…âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¥ğ‘‘ğ‘¡1/ğ‘ž1⋅⎧⎪⎨⎪⎩supp𝜕𝜓𝑅1ğœ•ğ‘¡ğ‘…ğœŽğ‘žâ€²1âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¥ğ‘‘ğ‘¡1/ğ‘žâ€²1⩽𝐶2⎧⎪⎨⎪⎩supp𝜕𝜓𝑅𝜕𝑡|𝑢|ğ‘ž1ğœ“ğ‘…âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¥ğ‘‘ğ‘¡1/ğ‘ž1⋅𝑅𝑁+ğœŽâˆ’ğœŽğ‘ž1/(ğ‘ž1−1)(ğ‘ž1−1)/ğ‘ž1,−𝑢𝐿∗𝜓𝑅𝑑𝑥𝑑𝑡⩽𝐶3Υ1|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž1⋅𝑅𝑁+ğœŽâˆ’â„“ğ‘ž1/(ğ‘ž1−1)(ğ‘ž1−1)/ğ‘ž1,(2.17) where Î¥1={(𝑥,𝑡)∶𝑡∈ℝ,𝐷𝛼𝑥𝜓𝑅(𝑥,𝑡)≠0forsome𝛼} and ğ‘žî…ž1=ğ‘ž1/(ğ‘ž1−1). It is essential here that the operator 𝐿∗ contains the derivatives of order ℓ only. It is obvious that supp𝜕𝜓𝑅𝜕𝑡∪Υ1⊂Σ≜{(𝑥,𝑡)∶𝑡∈ℝ,|𝑡|2+|𝑥|2ğœŽ>𝑅2ğœŽ}, and therefore inequality (2.15) implies that III⩽𝐶4Σ|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž1⋅𝑅(𝑁+ğœŽ)(ğ‘ž1−1)/ğ‘ž1î€·ğ‘…âˆ’ğœŽ+𝑅−ℓ(2.18)⩽𝐶4Σ|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž1𝑅𝐴(2.19)⩽𝐶4II1/ğ‘ž1𝑅𝐴(2.20) with 𝐴=(𝑁+ğœŽ)(ğ‘ž1−1)/ğ‘ž1−min{ğœŽ,ℓ}. Similarly, II⩽𝐶5Σ|𝑣|ğ‘ž2𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž2𝑅(𝑁+ğœŽ)(ğ‘ž2−1)/ğ‘ž2î€·ğ‘…âˆ’ğœŽ+ğ‘…âˆ’â„Žî€¸(2.21)⩽𝐶5Σ|𝑣|ğ‘ž2𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž2𝑅𝐵(2.22)⩽𝐶5III1/ğ‘ž2𝑅𝐵(2.23) with 𝐵=(𝑁+ğœŽ)(ğ‘ž2−1)/ğ‘ž2−min{ğœŽ,ℎ}. Then (2.20) and (2.23) lead to II(ğ‘ž1ğ‘ž2−1)/ğ‘ž1ğ‘ž2⩽𝐶6𝑅𝐵+𝐴1/ğ‘ž2,III(ğ‘ž1ğ‘ž2−1)/ğ‘ž1ğ‘ž2⩽𝐶7𝑅𝐴+𝐵1/ğ‘ž1.(2.24)
We consider the following cases. Case 1. 𝐵+𝐴(1/ğ‘ž2)<0. Let 𝑅→+∞ in the first inequality of (2.24), we obtain |𝑢|ğ‘ž1𝑑𝑥𝑑𝑡=0,(2.25) which implies 𝑢≡0. Combining with inequality (2.20) or equality (2.4), we get that ∫|𝑣|ğ‘ž2𝑑𝑥𝑑𝑡=0. Then 𝑣≡0.Case 2. 𝐴+𝐵(1/ğ‘ž1)<0. Inequality (2.24) implies 𝑣≡0. The inequality (2.23) or equality (2.5) leads that 𝑢≡0.Case 3. 𝐵+𝐴(1/ğ‘ž2)=0. By (2.22) and (2.19), we have II⩽𝐶8Σ|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž1ğ‘ž2𝑅𝐵+𝐴1/ğ‘ž2=𝐶8Σ|𝑢|ğ‘ž1𝜓𝑅𝑑𝑥𝑑𝑡1/ğ‘ž1ğ‘ž2.(2.26) And, from (2.24), we obtain that ∫|𝑢|ğ‘ž1𝑑𝑥𝑑𝑡 converge. Then, as 𝑅→+∞, Σ|𝑢|ğ‘ž1𝑑𝑥𝑑𝑡⟶0.(2.27) Then 𝑢≡0 and (2.20) implies 𝑣≡0.Case 4. 𝐴+𝐵(1/ğ‘ž1)=0. Similarly to Case 3, the second inequality of (2.24) implies ∫|𝑣|ğ‘žğ‘‘ğ‘¥ğ‘‘ğ‘¡ converging. 𝑣≡0 follows from (2.19) and (2.22). Then by (2.23) or (2.5), 𝑢≡0.
Taking ğœŽ=min{ℓ,ℎ}, Cases 1 and 3: 𝐵+𝐴(1/ğ‘ž2)⩽0 is equivalent to 𝑁/min{ℓ,ℎ}⩽(ğ‘ž1+1)/(ğ‘ž1ğ‘ž2−1), Cases 2 and 4: 𝐴+𝐵(1/ğ‘ž1)⩽0 is 𝑁/min{ℓ,ℎ}⩽(ğ‘ž2+1)/(ğ‘ž1ğ‘ž2−1). So the union of Cases 1–4 is just the set Γ1.
Similarly, taking ğœŽ=max{ℓ,ℎ}, we obtain that the union of Cases 1–4 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,ğ‘ž 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 sup𝑥∈ℝ𝑁||||𝑢(𝑥,𝑡)⩽𝐶(𝑇−𝑡)−(𝑝+1)/(ğ‘ğ‘žâˆ’1),sup𝑥∈ℝ𝑁||||𝑣(𝑥,𝑡)⩽𝐶(𝑇−𝑡)−(ğ‘ž+1)/(ğ‘ğ‘žâˆ’1)(3.1) for 𝑁2𝑚⩽max𝑝+1,ğ‘ğ‘žâˆ’1ğ‘ž+1î‚¼ğ‘ğ‘žâˆ’1.(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.

References

  1. P. Poláčik and P. Quittner, “Liouville type theorems and complete blow-up for indefinite superlinear parabolic equations,” in Nonlinear Elliptic and Parabolic Problems, vol. 64 of Progress in Nonlinear Differential Equations and Their Applications, pp. 391–402, Birkhäuser, Basel, The Switzerland, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. R. Xing, “The blow-up rate for positive solutions of indefinite parabolic problems and related Liouville type theorems,” Acta Mathematica Sinica (English Series), vol. 25, no. 3, pp. 503–518, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. H. Fujita, “On the blowing up of solutions of the Cauchy problem for ut=Δu+u1+α,” Journal of the Faculty of Science. University of Tokyo. Section IA, vol. 13, pp. 109–124, 1966. View at: Google Scholar
  4. D. G. Aronson and H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics,” Advances in Mathematics, vol. 30, no. 1, pp. 33–76, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,” Proceedings of the Japan Academy, vol. 49, pp. 503–505, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. K. Kobayashi, T. Sirao, and H. Tanaka, “On the growing up problem for semilinear heat equations,” Journal of the Mathematical Society of Japan, vol. 29, no. 3, pp. 407–424, 1977. View at: Publisher Site | Google Scholar
  7. F. B. Weissler, “Existence and nonexistence of global solutions for a semilinear heat equation,” Israel Journal of Mathematics, vol. 38, no. 1-2, pp. 29–40, 1981. View at: Publisher Site | Google Scholar
  8. K. Deng and H. A. Levine, “The role of critical exponents in blow-up theorems: the sequel,” Journal of Mathematical Analysis and Applications, vol. 243, no. 1, pp. 85–126, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems. Series A, vol. 8, no. 2, pp. 399–433, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. M. Escobedo and M. A. Herrero, “Boundedness and blow up for a semilinear reaction-diffusion system,” Journal of Differential Equations, vol. 89, no. 1, pp. 176–202, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. Y. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev, “On the necessary conditions of global existence to a quasilinear inequality in the half-space,” Comptes Rendus de l'Académie des Sciences. Série I, vol. 330, no. 2, pp. 93–98, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. P. Y. H. Pang, F. Sun, and M. Wang, “Existence and non-existence of global solutions for a higher-order semilinear parabolic system,” Indiana University Mathematics Journal, vol. 55, no. 3, pp. 1113–1134, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. È. Mitidieri and S. I. Pokhozhaev, “A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,” Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 234, pp. 1–384, 2001. View at: Google Scholar
  15. H. Pan and R. Xing, “Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 248–258, 2008. View at: Publisher Site | Google Scholar

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.

547 Views | 357 Downloads | 2 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder
 Sign up for content alertsSign up

You are browsing a BETA version of Hindawi.com. Click here to switch back to the original design.