Large Solutions for Semilinear Parabolic Equations Involving Some Special Classes of Nonlinearities

Abstract

We consider a new class of nonlinearities for which a nonlocal parabolic equation with Neumann boundary conditions has finite time blow-up solutions. Our approach is inspired by previous work done by Jazar and Kiwan (2008) and El Soufi et al. (2007).

1. Introduction

This paper is devoted to the existence of large solutions of the semilinear parabolic problem 𝑢_𝑡1−Δ𝑢=𝑓(|𝑢|)−𝑚(Ω)_Ω𝑓(|𝑢|)𝑑𝑥,inΩ,𝜕𝑢𝜕𝑛=0,on𝜕Ω, (1.1) with the initial conditions 𝑢(𝑥,0)=𝑢₀(𝑥),onΩ,where_Ω𝑢₀𝑑𝑥=0. (1.2) Here Ω⊂ℝ^𝑁 is a bounded regular domain of class 𝐶², 𝑓∶[0,∞)↦[0,∞) is a locally Lipschitz function, 𝑚(Ω) represents the Lebesgue measure of the domain Ω, and Δ is the Laplace operator.

The above problem was recently studied by El Soufi et al. [1] and Jazar and Kiwan [2], under the assumption that 𝑓 is a power function of the form 𝑓(𝑢)=𝑢^𝑝 (with 𝑝>1). Under the same restriction on 𝑓, some lower bounds estimates for the blow-up time were established in [3]. See also [4, 5].

The aim of our paper is to extend their results to a larger class of nonlinearities whose precise definition is as follows.

Definition 1.1. A real-valued function 𝑓 defined on an interval [𝑎,∞) (with 𝑎≥0) satisfies property (𝐶) if it is locally Lipschitz, nonnegative, and its mean value (1/(𝑡−𝑎))∫^𝑡_𝑎𝑓(𝑥)𝑑𝑥 has a superlinear growth in the sense that the ratio (1/(𝑡−𝑎))∫^𝑡_𝑎𝑓(𝑥)𝑑𝑥(𝑡−𝑎)^𝛼 (1.3) is nondecreasing for 𝑡 large enough and some 𝛼>1.

The monotonicity condition on (1.3) means precisely the existence of a constant 𝐶∈(0,1/2) (precisely, 𝐶=(1+𝛼)⁻¹) such that 1𝐶𝑓(𝑡)≥𝑡−𝑎^𝑡_𝑎𝑓(𝑥)𝑑𝑥 (1.4) for 𝑡>𝑎 large enough.

For example, if 𝑔∈𝐶¹([0,∞)), 𝑔(0)=0, and 𝑔 is nondecreasing, then the function 𝑓(𝑡)=𝑔(𝑡)𝑡^𝛼, with 𝛼>1, satisfies property (𝐶). In fact,^𝑡₀𝑡𝑓(𝑥)𝑑𝑥=^𝛼+1𝛼+1𝑔(𝑡)−^𝑡₀𝑔^′𝑥(𝑥)^𝛼+1≤𝑡𝛼+1𝑑𝑥_𝛼+1𝛼+1𝑔(𝑡)=𝐶𝑡𝑓(𝑡), (1.5) where 𝐶=1/(𝛼+1)∈(0,1/2).

Assuming that 𝑓(0)=0 (which is the case if 𝑎=0 and (1.4) works for all 𝑡>0), one can infer from (1.4) that𝑓(𝑡)+𝑓(0)2≥1𝑡^𝑡₀𝑓(𝑥)𝑑𝑥, (1.6) a fact that reminds of the Hermite-Hadamard inequality in convex functions theory. See [6, page 50]. Thus property (𝐶) can be ascribed to the field of generalized convexity.

The problems of type (1.1) and (1.2) arise naturally in mechanics, biology, and population dynamics. For example, if we consider a couple or a mixture of two equations of the above type, the resulting problem describes the temperatures of two substances, which constitute a combustible mixture, or represents a model for the behavior of densities of two diffusion biological species which interact with each other. This type of problems is connected also with parabolic systems of heat equations with local sources, which arise in population dynamics. See [4, 7–11].

Our paper is organized as follows. In Section 2 we show that every solution 𝑢 of the problems (1.1) and (1.2) (with 𝑢₀ not identically 0 and 𝑓 satisfying property (𝐶)) is large, provided that its energy at 𝑡=0 is nonpositive. See Theorem 2.4. Our approach combines previous work done by El Soufi et al. [1], with a careful analysis of the properties of energy of solutions.

In Section 3 we discuss the connection of property (𝐶) with other special classes of nonlinearities, well known in the literature. We prove that every function with generalized regular variation (à la Karamata), as well as every 𝑁-function in the sense of Orlicz, satisfies property (𝐶). Meantime property (𝐶) and the classical Keller-Osserman condition have a large overlap (though they are distinct from each other). Thus the class of functions satisfying property (𝐶) provides indeed a natural framework for the existence of large solutions for the problems (1.1) and (1.2).

2. The Existence of Large Solutions

The existence of a solution to the problems (1.1) and (1.2) can be found in [1]. It can be summarized as follows.

Theorem 2.1. Assume that Ω⊂ℝ^𝑁 is a bounded regular domain of class 𝐶² and 𝑓∶[0,∞)↦[0,∞) is a locally Lipschitz function. Then for every 𝑢₀∈𝐶(Ω) there is an element 𝑡_max∈[0,∞] such that the problems (1.1) and (1.2) has a unique solution 𝑢∈𝐶([0,𝑡_max);𝐶(Ω))∩𝐶¹((0,𝑡_max);𝐶(Ω)), (2.1) which solves the integral equation 𝑢(𝑡)=𝑒^𝑡𝛿𝑢₀+^𝑡₀𝑒^{(𝑡−𝑠)Δ}𝑓(𝑢(𝑠))𝑑𝑠 (2.2) on [0,𝑡_max). Moreover, _Ω𝑢(𝑡)𝑑𝑥=0,∀𝑡∈[0,𝑡_max), (2.3) and if 𝑡_max<∞, then lim_{𝑡→𝑡max}‖𝑢(𝑡)‖_𝐿^∞_(Ω)=∞.

Each solution 𝑢 of the problems (1.1) and (1.2) has the property ∫_Ω𝑢𝑑𝑥=0 because the integral in the right-hand side of (1.1) is 0 and 𝑑(𝑑𝑡_Ω𝑢𝑑𝑥)=_Ω𝑢_𝑡𝑑𝑥=_ΩΔ𝑢𝑑𝑥=0. (2.4) Hence, by the initial condition (1.2), we have ∫_Ω𝑢𝑑𝑥=0.

Lemma 2.2. Let 𝑢∈𝐶(Ω) be a solution of (1.1) and (1.2). Then the energy of 𝑢 at the moment 𝑡,𝐸(𝑡)=_Ω(12|∇𝑢|²−^𝑢₀𝑓(|𝑡|)𝑑𝑡)𝑑𝑥, (2.5) verifies the formula 𝐸(𝑡)=𝐸(0)−^𝑡_0Ω𝑢²_𝑡𝑑𝑥𝑑𝑡,∀𝑡>0. (2.6)

Proof. In fact, 𝑑𝐸𝑑𝑡(𝑡)=_Ω(∇𝑢_𝑡∇𝑢−𝑢_𝑡=𝑓(|𝑢|))𝑑𝑥_𝜕Ω𝜕𝑢𝑢𝜕𝑛^𝑡𝑑𝜎−_Ω𝑢_𝑡Δ𝑢𝑑𝑥−_Ω𝑢_𝑡𝑓(|𝑢|)𝑑𝑥=−_Ω𝑢_𝑡(Δ𝑢+𝑓(|𝑢|))𝑑𝑥=−_Ω𝑢²_𝑡𝑑𝑥, (2.7) and by integrating both sides over [0,𝑡], we obtain formula (2.6).

According to the previous lemma, if 𝐸(0) is nonpositive, then 𝐸(𝑡) is nonpositive for all 𝑡>0. In the case of functions 𝑓 satisfying condition (𝐶), this leads to 𝐶_Ω𝑢𝑓(|𝑢|)𝑑𝑥≥_Ω^𝑢₀1𝑓(|𝑡|)𝑑𝑡𝑑𝑥≥2_Ω|∇𝑢|². (2.8)

Lemma 2.3. Under the assumptions of Lemma 2.2 consider the two auxiliary functions 1𝑚(𝑡)∶=2_Ω𝑢²(𝑥,𝑡)𝑑𝑥,ℎ(𝑡)∶=^𝑡₀𝑚(𝑠)𝑑𝑠. (2.9) Then 𝑚^′1(𝑡)≥𝐶^𝑡_0Ω𝑢²_𝑡𝑑𝑡, (2.10)𝑚^′1(𝑡)≥(2𝐶−1)𝜆𝑚(𝑡),forsome𝜆>0, (2.11)12𝐶(ℎ^′(𝑡)−ℎ^′(0))²≤ℎ(𝑡)ℎ^′′(𝑡), (2.12) provided that 𝑓 satisfies condition (𝐶).

Proof. In fact, 𝑚^′(𝑡)=_Ω𝑢_𝑡𝑢𝑑𝑥=_Ω≥𝑢(Δ𝑢+𝑓(|𝑢|))𝑑𝑥_Ω(−|∇𝑢|²+1𝐶^𝑢₀1𝑓(|𝑡|)𝑑𝑡)𝑑𝑥=−𝐶_Ω(12|∇𝑢|²−^𝑢₀1𝑓(|𝑡|)𝑑𝑡)𝑑𝑥+(2𝐶−1)_Ω|∇𝑢|²𝑑𝑥. (2.13)
Hence, 𝑚^′1(𝑡)≥−𝐶1𝐸(𝑢)+(2𝐶−1)_Ω|∇𝑢|²1𝑑𝑥≥−𝐶1𝐸(𝑢)=−𝐶𝐸(𝑢₀1)+𝐶^𝑡_0Ω𝑢²_𝑡≥1𝑑𝑥𝑑𝑡𝐶^𝑡_0Ω𝑢²_𝑡𝑑𝑥𝑑𝑡. (2.14)
On the other hand, by the Poincaré inequality, we have 𝑚^′1(𝑡)≥(2𝐶−1)_Ω|∇𝑢|²1𝑑𝑥≥(2𝐶−1)𝜆_Ω𝑢²1𝑑𝑥=(2𝐶−1)𝜆𝑚(𝑡), (2.15) where 𝜆 is a suitable positive constant.
We pass now to the proof of (2.12). Since ℎ^′(𝑡)−ℎ^′(0)=^𝑡₀𝑚^′(𝑠)𝑑𝑠=^𝑡_0Ω𝑢𝑢_𝑡𝑑𝑥𝑑𝑡≤(^𝑡_0Ω𝑢²𝑑𝑥𝑑𝑡)^1/2(^𝑡_0Ω𝑢²_𝑡𝑑𝑥𝑑𝑡)^1/2≤(2ℎ(𝑡))^1/2(𝐶𝑚^′(𝑡))^1/2=(2𝐶ℎ(𝑡)ℎ^′′(𝑡))^1/2, (2.16) by (2.10) we infer that ℎ^′(𝑡)−ℎ^′(0)=^𝑡₀𝑚^′(𝑠)𝑑𝑠≥0, (2.17) and thus 12𝐶(ℎ^′(𝑡)−ℎ^′(0))²≤ℎ(𝑡)ℎ^′′(𝑡). (2.18)

We are now in a position to state the main result of our paper.

Theorem 2.4. Assume that 𝑓∶[0,∞)↦[0,∞) is a function with property (𝐶), and let 𝑢 be the solution of the problems (1.1) and (1.2) corresponding to an initial data 𝑢₀∈𝐶(Ω), 𝑢₀ not identically zero. If the energy of 𝑢 at 𝑡=0 is nonpositive, then 𝑢, as a function of 𝑡, cannot be in 𝐿^∞((0,𝑇);𝐿²(Ω)) for all 𝑇>0.

Proof. Suppose, by reduction ad absurdum, that the solution 𝑢(𝑥,⋅) exists in 𝐿^∞((0,𝑇);𝐿²(Ω)) (2.19) for all 𝑇>0. By (2.11), lim_𝑡→∞ℎ^′(𝑡)=lim_𝑡→∞𝑚(𝑡)=∞, (2.20) which yields, for each 𝛽∈(0,1/𝐶), the existence of a number 𝑇₀>0 such that for all 𝑡>𝑇₀, 𝛽ℎ^′(𝑡)²≤1𝐶(ℎ^′(𝑡)−ℎ^′(0))². (2.21) Now, by (2.12) we obtain 𝛽ℎ^′(𝑡)²≤2ℎ(𝑡)ℎ^′′(𝑡). (2.22)
We will show, by considering the function 𝐻(𝑡)=ℎ(𝑡)^−𝑞, for a suitable 𝑞>0, that the last inequality leads to a contradiction. In fact, 𝐻^′′(𝑡)=𝑞ℎ(𝑡)^−𝑞−2((𝑞+1)(ℎ^′(𝑡))²−ℎ(𝑡)ℎ^′′(𝑡))≤𝑞ℎ(𝑡)^−𝑞−2(2(𝑞+1)𝛽−1)ℎ(𝑡)ℎ^′′(𝑡), (2.23) for all 𝑡≥𝑇₀, so that for 𝛽∈(0,1/𝐶) and 𝑞∈(0,1/(2𝐶)−1) with 2(𝑞+1)<𝛽<1/𝐶, the corresponding function 𝐻(𝑡) is concave.
By (2.20), lim_𝑡→∞ℎ(𝑡)=∞, whence lim_𝑡→∞𝐻(𝑡)=0. Thus 𝐻 provides an example of a concave and strictly positive function which tends to 0 at infinity, a fact which is not possible. The proof is done.

3. Classes of Functions with Property (𝐶)

The aim of this section is to comment on how large is the class of functions which plays property (𝐶). In this respect we will discuss here several particular classes of functions with this property.

We start with the class of regularly varying functions, introduced by Karamata in [12].

Definition 3.1. A positive measurable function 𝑓 defined on interval [𝑎,∞) (with 𝑎≥0) is said to be regularly varying at infinity, of index 𝜎∈ℝ (abbreviated, 𝑓∈RV_∞(𝜎)), provided that lim_𝑥→∞𝑓(𝑡𝑥)𝑓(𝑥)=𝑡^𝜎,∀𝑡>0. (3.1)

All functions of index 𝜎 are of the form 𝑓(𝑥)=𝑥^𝜎exp(𝑎(𝑥)+^𝑥₀𝜀(𝑠)𝑠𝑑𝑠), (3.2) where 𝑎(𝑥) and 𝜀(𝑥) are bounded and measurable, 𝑎(𝑥)→𝛼∈ℝ, and 𝜀(𝑥)→0 as 𝑥→∞. In particular, so are𝑥^𝜎log𝑥,𝑥^𝜎loglog𝑥,𝑥^𝜎exp(log𝑥loglog𝑥),𝑥^𝜎exp((log𝑥)^1/3(cos(log𝑥)^1/3)). (3.3)

See [13] for details.

Semilinear problems with nonlinearities in the class of regularly varying functions have been studied by Cîrstea and Rădulescu [14].

Proposition 3.2. If 𝑓∈RV_∞(𝜎) with 𝜎>1, then lim_𝑥→∞𝐹(𝑥)=1𝑥𝑓(𝑥)<1𝜎+12, (3.4) where 𝐹(𝑥)∶=^𝑥₀𝑓(𝑠)𝑑𝑠. (3.5) Under these assumptions, 𝑓 satisfies condition (𝐶) (and thus Theorem 2.4 applies to it).

Proof. To prove this, consider the change of variable 𝑠=𝑡𝑥, which yields 𝐹(𝑥)=^𝑥₀𝑓(𝑠)𝑑𝑠=¹₀𝑥𝑓(𝑡𝑥)𝑑𝑡. (3.6)
The continuity of 𝑓 and the fact that 𝑓∈RV_∞(𝜎) assure the existence of a 𝛿>0 such that for every 𝑥>𝛿 we have 𝑓(𝑡𝑥)𝑓(𝑥)≤𝑡^𝜎+1, (3.7) whence the integrability of the function 𝑡→𝑓(𝑡𝑥)𝑓(𝑥) on [0,1]. Then lim_𝑥→∞𝐹(𝑥)𝑥𝑓(𝑥)=lim_𝑥→∞¹₀𝑓(𝑡𝑥)=𝑓(𝑥)𝑑𝑡₁₀lim_𝑥→∞𝑓(𝑡𝑥)𝑓(𝑥)𝑑𝑡=¹₀𝑡^𝜎1𝑑𝑡=,𝜎+1 (3.8) where the commutation of the limit with the integral is motivated by the Lebesgue dominated convergence theorem.

An important class of nonlinearities which appeared in connection with the study of boundary blow-up problems for elliptic equations is that of functions satisfying the Keller-Osserman condition. See the papers by Rădulescu [15] and Dumont et al. [16].

Definition 3.3. A nonnegative and nondecreasing function 𝑓∈𝐶¹([0,∞)) with 𝑓(0)=0 satisfies the generalized Keller-Osserman condition of order 𝑝>1 if ^∞₁1(𝐹(𝑡))^1/𝑝𝑑𝑡<∞, (3.9) where 𝐹 is the primitive of 𝑓 given by formula (3.5).

If 𝑓∈RV_∞(𝜎+1) with 𝜎+2>𝑝>1 a nondecreasing and continuous function, then 𝐹∈RV_∞(𝜎+2) and 𝐹^−1/𝑝∈RV_∞((−𝜎−2)/𝑝). Since (−𝜎−2)/𝑝<−1, we infer that 𝐹^−1/𝑝∈𝐿¹([1,∞)) and thus 𝑓 satisfies the generalized Keller-Osserman condition.

It is worth to notice that the function exp(𝑡) is not regularly varying at infinity though satisfies the generalized Keller-Osserman condition and also the hypothesis of Proposition 3.4.

Necessarily, if a function 𝑓 satisfies the generalized Keller-Osserman condition of order 𝑝>1, then lim_𝑡→∞𝐹(𝑡)𝑡^𝑝=∞, (3.10) while 𝐹(𝑡)/𝑡^𝑝 may be (or may be not) a monotonic function.

As noticed in the Introduction, property (𝐶) is intimately related to the monotonicity of 𝐹(𝑡)/𝑡^𝑝 in the following way.

Proposition 3.4. If 𝐹(𝑡)/𝑡^𝑝 is nondecreasing for some 𝑝>2, then the function 𝑓 satisfies condition (𝐶) with 𝐶=1/𝑝 (and thus Theorem 2.4 applies to it).

According to Proposition 3.4, the function 𝑓(𝑡)=𝑝𝑡^𝑝−1log(𝑡+1)+(𝑡^𝑝/(𝑡+1)) satisfies for 𝑝>2 condition (𝐶)   but not the generalized Keller-Osserman condition of order 𝑝. Indeed, 𝑓 admits the primitive 𝐹(𝑡)=𝑡^𝑝log(𝑡+1).

We end our paper by discussing the connection of property (𝐶) with a class of functions due to Orlicz.

Definition 3.5. An 𝑁-function is any function 𝑀∶[0,∞)→ℝ of the form 𝑀(𝑥)=^𝑥₀𝑝(𝑡)𝑑𝑡, (3.11) where 𝑝 is nondecreasing and right continuous, 𝑝(0)=0,  𝑝(𝑡)>0 for 𝑡>0, and lim_𝑡→∞𝑝(𝑡)=∞.
An 𝑁-function 𝑀 satisfies the Δ₂-condition if there exist constants 𝑘>0 and 𝑥₀≥0 such that 𝑀(2𝑥)≤𝑘𝑀(𝑥),∀𝑥≥𝑥₀. (3.12)

Any 𝑁-function 𝑀 is convex and plays the following properties:(N1)𝑀(0)=0 and 𝑀(𝑥)>0 for 𝑥>0;(N2)𝑀(𝑥)/𝑥→0 as 𝑥→0 and 𝑀(𝑥)/𝑥→∞ as 𝑥→∞.

Two examples of 𝑁-functions which satisfy the Δ₂-condition are 𝑥^𝑝/𝑝 (for 𝑝≥1) and 𝑡(log𝑡)⁺.

The 𝑁-functions which satisfy the Δ₂-condition are instrumental in the theory of Orlicz spaces (which extend the 𝐿^𝑝(𝜇) spaces). Their theory is available in many books, such as [17, 18], and has important applications to interpolation theory [19] and Fourier analysis [20].

According to [18, page 23], the constant 𝑘 which appears in the formulation of Δ₂-condition is always greater than or equal to 2.

Proposition 3.6. Every 𝑁-function 𝑀∶[0,∞)→ℝ which satisfies the Δ₂-condition has property (𝐶) (and thus Theorem 2.4 applies to it).

Proof. Since 𝑀 is nondecreasing, 𝑀(𝑡𝑥)=𝑀(2^log2𝑡𝑥)≤𝑀(2^[log2𝑡]+1𝑥), (3.13) and taking into account the Δ₂-condition we infer that 𝑀(𝑡𝑥)≤𝑀(𝑥)𝑘^[log2𝑡]+1≤𝑀(𝑥)𝑘^log2𝑡+1≤𝑀(𝑥)𝑡^2log2𝑘, (3.14) for all 𝑥  big enough and 𝑡≥2. Hence, ^𝑡₀𝑀(𝑥)𝑑𝑥=¹₀≤𝑡𝑀(𝑡𝑠)𝑑𝑠₁₀𝑡𝑀(𝑡)𝑠^2log2𝑘1𝑑𝑠=2log₂≤1𝑘+1𝑡𝑀(𝑡)3𝑡𝑀(𝑡), (3.15) and the proof is done.

Acknowledgment

The authors have been supported by CNCSIS Grant 420/2008.