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

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 𝑢_𝑡+𝑢((𝑥−,Δ0))^𝑚=𝑢𝑢=₀|𝑣|^𝑝,𝑣_𝑡+(−Δ)^𝑚𝑣=|𝑢|^𝑞,(𝑥,𝑡)∈ℝ^𝑁×(0,𝑇),(𝑥)∈𝐿^∞ℝ^𝑁,𝑣(𝑥,0)=𝑣₀(𝑥)∈𝐿^∞ℝ^𝑁,(1.1) where 𝑚⩾1 and 𝑝,𝑞>1; the other is parabolic Liouville theorems for the problem 𝑢_𝑡+(−𝑢Δ()𝑥^𝑚,𝑢𝑡)=∈|𝑣𝐿|^𝑝,_𝑝loc𝑣_𝑡+(−Δ)^𝑚𝑣=|𝑢|^𝑞,(𝑥,𝑡)∈ℝ^𝑁×ℝ,ℝ^𝑁+1,𝑣(𝑥,𝑡)∈𝐿^𝑞locℝ^𝑁+1,(1.2) where 𝑚⩾1 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∈ℝ (see, e.g., Poláčik and Quittner [1] and Xing [2]).

Recall that, in his famous paper [3], Fujita studied the initial value problem 𝑢_𝑡−Δ𝑢𝑢(=𝑥,𝑢0^𝑝),=𝑢₀(𝑥,𝑡)∈ℝ^𝑁×(0,∞),(𝑥),𝑥∈ℝ^𝑁,(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, ℝ^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 [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 𝑢(𝑥,0)𝑢=_𝑡−𝑢₀Δ𝑢=𝑣^𝑝,𝑣_𝑡−Δ𝑣=𝑢^𝑞,(𝑥,𝑡)∈ℝ^𝑁×(0,𝑇),(𝑥)⩾0∈𝐿^∞ℝ^𝑁,𝑣(𝑥,0)=𝑣₀(𝑥)⩾0∈𝐿^∞ℝ^𝑁,(1.4) with 𝑝>0 and 𝑞>0. They established the following.(i)If 0<𝑝𝑞⩽1, then all solutions are global.(ii)If 𝑝𝑞>1 and 𝑁/2⩽max{(𝑝+1)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−1)}, then every nontrivial solution blows up in finite time.(iii)If 𝑝𝑞>1 and 𝑁/2>max{(𝑝+1)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−1)}, then every solution with initial data having positive average value does not exist globally in time.(ii)If 𝑁/2𝑚>max{(𝑝+1)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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 𝑞>1. Then   (i)if 𝑁/2𝑚⩽max{(𝑝+1)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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{(𝑞+1)/(𝑝𝑞−1),(𝑝+1)/(𝑝𝑞−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 sup_𝑥∈ℝ^𝑁|𝑢(𝑥,𝑡)|,sup_𝑥∈ℝ^𝑁|𝑣(𝑥,𝑡)|⩽𝐶(𝑇−𝑡)^{−(𝑞+1)/(𝑝𝑞−1)} for 𝑁/2𝑚⩽min{(𝑝+1)/(𝑝𝑞−1),(𝑞+1)/(𝑝𝑞−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{(𝑞+1)/(𝑝𝑞−1),(𝑝+1)/(𝑝𝑞−1)} to 𝑁/2𝑚⩽max{(𝑞+1)/(𝑝𝑞−1),(𝑝+1)/(𝑝𝑞−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,𝐷_𝑥) be a differential operator of order ℓ: 𝐿[𝑣]∶=_|𝛼|=ℓ𝐷^𝛼𝑎_𝛼(𝑡,𝑥,𝑣)𝑣,(2.1) and let 𝑀 be a differential operator of order ℎ: 𝑀[𝑣]∶=_||𝛽||=ℎ𝐷^𝛽𝑏_𝛽(𝑡,𝑥,𝑣)𝑣,(2.2) where 𝑎_𝛼(𝑡,𝑥,𝑣) and 𝑏_𝛽(𝑡,𝑥,𝑣) are bounded functions defined for 𝑡∈ℝ,𝑥∈ℝ^𝑁,and𝑣∈ℝ.

Consider the set of (𝑢,𝑣) satisfying the inequalities: 𝜕𝑢𝜕𝑡⩾𝐿[𝑢]+|𝑣|^𝑞2,𝜕𝑣𝜕𝑡⩾𝑀[𝑣]+|𝑢|^𝑞1,(𝑥,𝑡)∈ℝ^𝑁×ℝ,(2.3)

in the following weak sense: if 𝜓∈𝐶^{0max{ℓ,ℎ}}(ℝ^𝑁+1) and 𝜓(𝑥,𝑡)⩾0, then −𝜕𝜓𝜕𝑡𝑢𝑑𝑥𝑑𝑡−𝑢𝐿^∗−𝜕𝜓[𝜓]𝑑𝑥𝑑𝑡⩾|𝑣|^𝑞2𝜓𝑑𝑥𝑑𝑡,(2.4)𝜕𝑡𝑣𝑑𝑥𝑑𝑡−𝑣𝑀^∗[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 ℝ^𝑁×ℝ, and 𝐿^∗[𝜓]∶=_|𝛼|=ℓ𝑎_𝛼(𝑡,𝑥,𝑢)(−𝐷)^𝛼𝜓,𝑀^∗[𝜓]∶=_|𝛼|=ℎ𝑏_𝛼(𝑡,𝑥,𝑢)(−𝐷)^𝛼𝜓.(2.6)

Here is the main result of this section.

Theorem 2.1. If two functions 𝑢(𝑥,𝑡)∈𝐿^𝑞1_loc(ℝ^𝑁+1) and 𝑣(𝑥,𝑡)∈𝐿^𝑞2_loc(ℝ^𝑁+1) satisfy (2.4) and (2.5), then 𝑢(𝑥,𝑡)≡0,𝑣(𝑥,𝑡)≡0 for 𝑞₁,𝑞₂>1 and (𝑞₁,𝑞₂)∈Γ₁∪Γ₂, where Γ₁=𝑞₁,𝑞₂∣𝑁min{ℓ,ℎ}⩽max𝑞¹+1𝑞₁𝑞₂−1,𝑞²+1Γ𝑞₁𝑞₂−1₂,=𝑞₁,𝑞₂∣𝑁+max{ℓ,ℎ}⩽maxℎ+𝑞¹ℓ+ℎ𝑞₁𝑞₂−1,ℓ+𝑞²ℎ+ℓ𝑞₁𝑞₂−1.(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,𝑞₂∣𝑁min{ℓ,ℎ}⩽min𝑞¹+1𝑞₁𝑞₂−1,𝑞²+1𝑞₁𝑞₂−1(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|^𝑞.

Corollary 2.3. If two functions 𝑢(𝑥,𝑡)∈𝐿^𝑝loc(ℝ^𝑁+1) and 𝑣(𝑥,𝑡)∈𝐿^q_loc(ℝ^𝑁+1) satisfy 𝑢_𝑡+(−Δ)^𝑚𝑢=|𝑣|^𝑝,𝑣_𝑡+(−Δ)^𝑚𝑣=|𝑢|^𝑞,(2.9) on ℝ^𝑁×ℝ, then 𝑢(𝑥,𝑡)≡0,𝑣(𝑥,𝑡)≡0 for 𝑝,𝑞>1 belonging to the following set: (𝑝,𝑞)∣2𝑁𝑚⩽max𝑝𝑞𝑞+−11,𝑝𝑝𝑞+−11.(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∈𝐶^{0max{ℓ,ℎ}}(ℝ),𝜙⩾0, and 𝜙(𝑠)=01aass𝑠𝑠⩾⩽12,.(2.11) Suppose that there exists a positive constant 𝐶 such that ||||𝜙𝜙^||(𝑠)⩽𝐶𝜙^1/𝑞1(𝑠),||𝜙^(ℓ)||(𝑠)⩽𝐶𝜙^1/𝑞1(𝑠),_||(𝑠)⩽𝐶𝜙^1/𝑞2(𝑠),||𝜙^(ℎ)||(𝑠)⩽𝐶𝜙^1/𝑞2(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{ℓ𝑞₁/(𝑞₁−1),ℎ𝑞₂/(𝑞₂−1)}.
Let 𝜓_𝑅(𝑥,𝑡)=𝜙|𝑡|²+|𝑥|^2𝜎𝑅^2𝜎,𝑅>0,(2.13) the value of the parameter 𝜎>0 will be determined below. Now putting 𝜓=𝜓_𝑅(𝑥,𝑡) in (2.4) and (2.5) and letting II=|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡,III=|𝑣|^𝑞2𝜓_𝑅𝑑𝑥𝑑𝑡,(2.14) we have III⩽−𝜕𝜓^𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡−𝑢𝐿^∗𝜓_𝑅𝑑𝑥𝑑𝑡,(2.15)II⩽−𝜕𝜓^𝑅𝜕𝑡𝑣𝑑𝑥𝑑𝑡−𝑣𝑀^∗𝜓_𝑅𝑑𝑥𝑑𝑡.(2.16)
The Hölder inequality implies −𝜕𝜓^𝑅𝜕𝑡𝑢𝑑𝑥𝑑𝑡⩽𝐶⁰_supp𝜕𝜓_𝑅𝜕𝑡|𝑢|𝜓^1/𝑞1𝑅|𝑡|𝑅^2𝜎⩽𝐶₁𝑑𝑥𝑑𝑡⎧⎪⎨⎪⎩_supp𝜕𝜓_𝑅𝜕𝑡|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡⎫⎪⎬⎪⎭^1/𝑞1⋅⎧⎪⎨⎪⎩_supp𝜕𝜓_𝑅𝜕𝑡1𝑅^𝜎𝑞′1𝑑𝑥𝑑𝑡⎫⎪⎬⎪⎭^1/𝑞′1⩽𝐶₂⎧⎪⎨⎪⎩_supp𝜕𝜓_𝑅𝜕𝑡|𝑢|^𝑞1−𝑢𝐿_∗𝜓_𝑅𝑑𝑥𝑑𝑡⎫⎪⎬⎪⎭^1/𝑞1⋅𝑅^{𝑁+𝜎−𝜎𝑞1/(𝑞1−1)}^{(𝑞1−1)/𝑞1},𝜓_𝑅𝑑𝑥𝑑𝑡⩽𝐶₃_Υ1|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞1⋅𝑅^{𝑁+𝜎−ℓ𝑞1/(𝑞1−1)}^{(𝑞1−1)/𝑞1},(2.17) where Υ₁={(𝑥,𝑡)∶𝑡∈ℝ,𝐷^𝛼𝑥𝜓_𝑅(𝑥,𝑡)≠0forsome𝛼} and 𝑞^1=𝑞₁/(𝑞₁−1). It is essential here that the operator 𝐿^∗ contains the derivatives of order ℓ only. It is obvious that supp𝜕𝜕𝜓𝑡^𝑅∪Υ¹⊂Σ≜{(𝑥,𝑡)∶𝑡∈ℝ,|𝑡|²+|𝑥|^2𝜎>𝑅^2𝜎}, and therefore inequality (2.15) implies that III⩽𝐶₄_Σ⩽𝐶₄|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞1⋅𝑅^{(𝑁+𝜎)(𝑞1−1)/𝑞1}𝑅^−𝜎+𝑅^−ℓ(2.18)_Σ⩽𝐶₄|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞1𝑅^𝐴(2.19)II^1/𝑞1𝑅^𝐴(2.20) with 𝐴=(𝑁+𝜎)(𝑞₁−1)/𝑞₁−min{𝜎,ℓ}. Similarly, II⩽𝐶₅_Σ⩽𝐶₅|𝑣|^𝑞2𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞2𝑅^{(𝑁+𝜎)(𝑞2−1)/𝑞2}𝑅^−𝜎+𝑅^−ℎ(2.21)_Σ⩽𝐶₅|𝑣|^𝑞2𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞2𝑅^𝐵(2.22)III^1/𝑞2𝑅^𝐵(2.23) with 𝐵=(𝑁+𝜎)(𝑞₂−1)/𝑞₂−min{𝜎,ℎ}. Then (2.20) and (2.23) lead to II^{(𝑞1𝑞2−1)/𝑞1𝑞2}⩽𝐶₆𝑅^{𝐵+𝐴1/𝑞2},III^{(𝑞1𝑞2−1)/𝑞1𝑞2}⩽𝐶₇𝑅^{𝐴+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 𝑢≡0. Combining with inequality (2.20) or equality (2.4), we get that ∫|𝑣|^𝑞2𝑑𝑥𝑑𝑡=0. Then 𝑣≡0.Case 2. 𝐴+𝐵(1/𝑞₁)<0. Inequality (2.24) implies 𝑣≡0. The inequality (2.23) or equality (2.5) leads that 𝑢≡0.Case 3. 𝐵+𝐴(1/𝑞₂)=0. By (2.22) and (2.19), we have II⩽𝐶₈_Σ|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞1𝑞2𝑅^{𝐵+𝐴1/𝑞2}=𝐶₈_Σ|𝑢|^𝑞1𝜓_𝑅𝑑𝑥𝑑𝑡^1/𝑞1𝑞2.(2.26) And, from (2.24), we obtain that ∫|𝑢|^𝑞1𝑑𝑥𝑑𝑡 converge. Then, as 𝑅→+∞, _Σ|𝑢|^𝑞1𝑑𝑥𝑑𝑡⟶0.(2.27) Then 𝑢≡0 and (2.20) implies 𝑣≡0.Case 4. 𝐴+𝐵(1/𝑞₁)=0. Similarly to Case 3, the second inequality of (2.24) implies ∫|𝑣|^𝑞𝑑𝑥𝑑𝑡 converging. 𝑣≡0 follows from (2.19) and (2.22). Then by (2.23) or (2.5), 𝑢≡0.
Taking 𝜎=min{ℓ,ℎ}, Cases 1 and 3: 𝐵+𝐴(1/𝑞₂)⩽0 is equivalent to 𝑁/min{ℓ,ℎ}⩽(𝑞₁+1)/(𝑞₁𝑞₂−1), Cases 2 and 4: 𝐴+𝐵(1/𝑞₁)⩽0 is 𝑁/min{ℓ,ℎ}⩽(𝑞₂+1)/(𝑞₁𝑞₂−1). So the union of Cases 1–4 is just the set Γ₁.
Similarly, taking 𝜎=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,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 sup_{𝑥∈ℝ𝑁}||𝑢(𝑥,𝑡)||⩽𝐶(𝑇−𝑡)^{−(𝑝+1)/(𝑝𝑞−1)},sup_{𝑥∈ℝ𝑁}||𝑣(𝑥,𝑡)||⩽𝐶(𝑇−𝑡)^{−(𝑞+1)/(𝑝𝑞−1)}(3.1) for 𝑁2𝑚⩽max𝑝𝑝𝑞+−11,𝑝𝑞𝑞+−11.(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.