Global Existence and Blow-up of Solutions to a Quasilinear Parabolic Equation with Nonlocal Source and Nonlinear Boundary Condition

This paper investigates the behavior of positive solution to the following p-Laplacian equation ut − (|ux | ux )x = ∫ a 0 u(ξ, t)dξ+ ku (x , t), (x , t) ∈ [0, a]× (0, T) with nonlinear boundary condition ux |x=0 = 0, ux |x=a = u q |x=a, where p ≥ 2, α,β , k, q > 0. The authors first get the local existence result by a regularization method. Then under appropriate hypotheses, the authors establish that positive weak solution either exists globally or blow up in finite time by using comparison principle.

Over the last few years, much effort has been devoted to the study of blow-up properties for nonlocal semilinear parabolic equations with nonlinear boundary conditions.Conditions on blowing up, blow-up set, blow-up rate and asymptotic behavior of solutions are obtained (see [10,11,12,21,22,23,24] and the references therein).The problem concerning (1.1) include the existence and multiplicity of global solutions, blowing-up, blow-up rates and blow-up sets, uniqueness and nonuniqueness etc.
Peng and Yang [12] investigated the blow-up properties of the following problem The motivation for studying problem (1.2) comes from Ockendon's model (see [13]) for the flow in a channel of a fluid whose viscosity depends on temperature where u represents the temperature of the fluid.
In [14], Galaktionov and Levine studied the heat conduction equation with gradient dependent diffusion x > 0, (1.4) where m ≥ 1 and u 0 has compact support.They proved that the critical global exponent and the critical Fujita exponent for the problem (1.4).
For the p-Laplacian equation, a few authors (see [15,16]) have investigate the following equation: (1.5) with initial and boundary conditions.Roughly speaking, their results are: (1) the solution u exists globally if q < p − 1, and (2) u blows up in finite time if q > p − 1 and u 0 (x) is sufficiently large.
The authors in [17] studied the following equation: with null Dirichlet conditions and obtain that the solution either exists globally or blows up in finite time.Under appropriate hypotheses, they have local theory of the solution and obtain that the solution either exists globally or blow-up in finite time.
Recently, in [18], the following problem has been intensively studied by X. Wu: where T > 0, m 1 , m 2 , q 11 , q 12 , q 21 , q 22 > 0, p 1 , p 2 > 1.They proved the global existence and blow-up of positive weak solutions of (1.7) by using comparison principle.However, to the author ąŕs best knowledge, there is little literature on the study of the global existence and blow-up properties for the system (1.1).Motivated by the above works, in this paper, we study global existence or blow-up of weak solutions to (1.1).Note that (1.1) has nonlinear and nonlocal source a 0 u α (ξ, t)dξ, local term u β (x, t) and nonlinear boundary condition u q , which make the behavior of the solution different from that for that of homogeneous Neumann or Dirichlet boundary value problems.To overcome these difficulties, we used some modification of the technique in [19] so that we can handle the nonlinearities.In this paper, the blow-up means that there exists a T * < +∞ such that u(•, t) ∞ < ∞ for t ∈ (0, T * ) and lim t→T * u(•, t) ∞ = ∞.The outline of this paper is as follows: in the next section, we will prove the local existence results by a regularization method.We will give the proof of a weak comparison principle and discuss the global existence and blow-up of solutions in the third section.

Local existence
In this section, we study the local existence of (1.1) under appropriate hypotheses.From the point of physics, we need only to consider the nonnegative solutions.Since (1.1) is the degenerate parabolic equations for |u x | = 0, one cannot expect the existence of classical solution.As it is now well known that degenerate equations need not posses classical solutions, most of studies of p-Laplacian equations concerned with weak solutions (see [7,9]).We begin by giving a precise definition of a weak solution for problem (1.1).
Before stating our main results, we make some assumptions.Let D = (0, a) and Ω r = D × (0, r].Let D and Ω r be their respective closures, z + = max{z, 0}.Before stating our main results, we make some assumptions on the initial value u 0 (x). (2.1) A weak solution of (1.1) is a vector function which is both a subsolution and a supersolution of (1.1).For every T < ∞, if u is a solution of (1.1), we say u is global.
In this case, we say that the solution u blows up in finite time.
Proof.The proof of this theorem basically follows line by line the proof of Theorem 1 in [19].Consider the following approximate problems for (1.1): (2.2) We need to control the nonlocal term by applying the technique developed in [19].Choose the bounded functions: And we assume that there exist positive constants l and L such that 2) are a nondegenerate problem for each fixed ǫ.We divide our proof into four steps.
Step 1.There exists a small constant t 1 > 0 and a positive constant C independent of ǫ such that: (

2.3)
To this end, choose bounded functions: where K will be determined later.Then consider the following problem: (2.4) For (2.4), standard parabolic theory (see [20]) shows that there is a solution u ǫ in the class H 2+β,1+β/2 (Ω T ) for some β ∈ (0, 1).Obviously, Comparison principle holds for (2.4).Therefore, Similarly to the proof of Proposition 3.1 in [19], there exists a small constant t 1 > 0 such that: where we get the conclusion.
We may assume that T ∈ [0, 1).Let h ≥ M .In fact, multiplying (2.2) by (u ǫ −h) + and integrating over Ω T , we obtain a 0 for some positive constant c independent of ǫ.Then similarly to the proof of Proposition 3.1 in [19], there exists a t 2 > 0, independent of ǫ, such that Step 3.There exist constants M 1 , independent of ǫ, such that To do so, multiplying (2.2) by u ǫt and integrating over Ω T , we have Similarly to the proof of Proposition 3.1 in [19], we have Using Young ′ s inequality, we have By the Gronwall ′ s lemma, we obtain the desired results.
Step 4. Therefore, by the Aubin theorem, it follows that (up to extraction of a subsequence): From (2.3), we have Then similarly to the proof of Theorem 2.1 in [15], we have The proof of Theorem 2.3 is completed by a standard limiting process.

Global Existence and Blow-up
In this section, we shall discuss the global existence and blow-up in finite time of the solution for system (1.1).Throughout this section we denote τ = p−1 p and choose λ, λ satisfying λ > 1 > λ > 0.
Our approach in a combination principle and upper and sub-technique.In order to prove our results, we give the following weak comparison principle.
Proof.It is easy to prove that there exist positive constants m 1 > 0, m 2 > 1 satisfying Set Computing directly, we obtain By (3.2) and (3.3) we have For the conditions defined before, we can also get also we have Then there exists constants a 1 > 0 such that the solution u(x, t) of (1.1) exists globally when u 0 (x) ≤ a 1 ϕ(x).