- Open Access
Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition
© Fang and Zhang; licensee Springer. 2014
- Received: 13 September 2013
- Accepted: 11 December 2013
- Published: 7 January 2014
In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.
- p-Laplacian evolution equation
- nonlinear memory
- global existence
- weight function
under the weighted linear nonlocal Dirichlet boundary condition (1.1). By virtue of the method of an upper-lower solution, they obtained global existence, blow-up properties, and blow-up rate of solutions.
for the special case and . It is necessary to point out that assumption (1.2) seems to be reasonable, but unfortunately, the authors of  did not give a relationship between and equation (1.2). The characterization of the monotonicity condition (1.2) was given by Souplet in , who proved the existence of monotone in time solutions for the above problem and obtained the blow-up rate (1.4) without the assumption of condition (1.3).
where , , . They got similar results by the method of upper-lower solution. We should notice that this kind of equation can be turned into a degenerate porous medium equation by suitable transformation. In addition, for the system of porous medium equations with nonlinear memory terms and a homogeneous Dirichlet boundary condition, one can refer for example to [8, 9].
where . They gave the conditions of global existence and blow-up of solutions and the blow-up rate of solutions for , by establishing an auxiliary function.
where , , , , , and () is a bounded domain with smooth boundary. The weight function in the boundary condition is continuous, nonnegative on , and on ∂ Ω, while the nonnegative and nontrivial initial data satisfies the compatibility conditions for and for , which is the closed relationship for local solvability of our problem (1.5)-(1.7) (see Section 2).
The nonlocal diffusion model like equation (1.5) arises in many natural phenomena. In some sense, this kind of nonlocal problem is closer to the actual model than the local problem, such as the model of non-Newton flux through a porous medium, the model for compressible reactive gases, the model of population dynamics, and the model of biological species with human-controlled distribution (see [2, 11–14] and references therein). From a physics point of view, equation (1.5) with , and appears in the theory of nuclear reactor dynamics in which case the nonlocal term with time-integral is called the memory term . In fact, there are some important phenomena formulated as parabolic equations which are coupled with weighted nonlocal boundary conditions in mathematical models, such as thermoelasticity theory. In this case, the solution describes the entropy per volume of the materia1 (see [16, 17]).
Our main goal is to find the effects of weight function on global or non-global existence of solutions for problem (1.5)-(1.7), the suitable range of nonlinear exponent, and to give the blow-up rate estimate under some suitable conditions. In addition, we treat the nonlocal nonlinearity Hölder (non-Lipschitz) cases m or , as well as the Lipschitz cases in this paper. We get our main results by establishing a modified comparison principle, constructing the suitable upper and lower solutions (including the self-similar lower solutions, the eigenfunction argument and the technique of ordinary differential equation and so on) and the auxiliary function. Moreover, our results extend part of or all results in [8–10]. The detailed results are stated as follows.
For arbitrary . If , then the solution of problem (1.5)-(1.7) blows up in finite time for sufficiently large initial data.
If , for . If , then the solution of problem (1.5)-(1.7) blows up in finite time for all strictly positive initial dates with T sufficiently large.
If , for . , , the initial value satisfies conditions (H1)-(H2) (see Section 3) and is the blow-up solution of problem (1.2)-(1.4), then the blow-up rate is
where , and is a constant.
If , for ,
If , then the solution of problem (1.5)-(1.7) exists globally for small initial data.
If , and , then the solution of problem (1.5)-(1.7) exists globally for small initial data.
If , then the solution of problem (1.5)-(1.7) exists globally for small enough initial data.
The rest of the paper is organized as follows. In Section 2, we give the preliminaries for our research. The proofs of blow-up results and blow-up rate of solutions are given in Section 3. In Section 4, we will deduce the results of global existence.
Since equation (1.2) is degenerate when , there is no classical solution in general. Hence, it is reasonable to find a weak solution. To this end, we first give the following definition of nonnegative weak solution of problem (1.5)-(1.7).
then is called the weak solution of problem (1.5)-(1.7).
If the equalities in equations (2.1)-(2.3) are replaced by ‘≤’ and ‘≥’, we can get and which are called the lower solution and upper solution of problem (1.5)-(1.7), respectively.
The following modified comparison principle plays a crucial role in our proofs, which can be obtained by establishing a suitable test function and Gronwall’s inequality.
Proposition 1 (Comparison principle)
Suppose that and are the lower and upper solutions of problem (1.2)-(1.4), respectively. If , and , where ε is any positive constant, then in .
By Gronwall’s inequality, we can deduce that , and so in .
in the case of in Ω. Therefore, we obtain on , and in . □
Next, we state the theorem of local existence and uniqueness without proof.
Theorem (Local existence and uniqueness)
Remark 1 The existence of local nonnegative solutions in time to problem (1.5)-(1.7) can be obtained by combining Theorem 1.2 in  with Theorem A4′ in . By the comparison principle above, we can get the uniqueness of the solutions to problem (1.5)-(1.7) with , .
Comparing with the problem under a general homogeneous Dirichlet boundary condition, the existence of weight function in the boundary condition has a great influence on the global and non-global existence of solutions.
Theorem 1 Suppose that , then the solution of problem (1.5)-(1.7) blows up in finite time for arbitrary and sufficiently large initial data.
We will discuss the problem for two cases.
It is obvious that it holds for .
- (i); we have , then
in the sense of and .
- (ii); we have and . Since and , we know that , and(3.2)
choose , then is the lower solution of problem (1.5)-(1.7). This implies that the solution blows up in finite time for large enough initial data. □
Theorem 2 Suppose that for . If , then the solution of problem (1.5)-(1.7) blows up in finite time for all strictly positive initial dates with T sufficiently large.
Since and , the solution of this problem blows up in finite time if .
It is obvious that the solution of problem (3.3) is a lower solution of problem (1.5)-(1.7) when and . By Proposition 1, is a blow-up solution. □
Suppose that the solution of problem (1.5)-(1.7) with blows up in finite time, and let . We suppose that the initial data satisfies the following assumptions:
(H2) There exists a constant such that .
where and .
Remark 2 Choose , , , and , one can easily verify that satisfies (C1)-(C2), conditions in Theorem 3 are thus valid.
Lemma 1 If satisfies condition (H1)-(H2), , then there exists a positive constant such that .
Integrating the result above over , we can obtain the conclusion. □
It follows from the assumptions of (H1)-(H2) that . Therefore, it is easy to deduce that for . That is, and integrating this over yields . Combining the results with Lemma 1, we obtain the desired result. □
In this section, we give sufficient conditions of the global existence of solutions.
Theorem 4 Suppose that for . If , then the solution of problem (1.5)-(1.7) exists globally for small initial data.
Selecting , we can deduce that the result holds. □
Theorem 5 Suppose that for . If , and , then the solution of problem (1.5)-(1.7) exists globally for small initial data.
in which . Let .
Choosing to be sufficiently small such that , we can conclude that is an upper solution of problem (1.5)-(1.7). The proof is completed. □
Theorem 6 Suppose that for . If , then the solution of problem (1.5)-(1.7) exists globally for small enough initial data.
Proof Choosing and , it is easy to see that the result holds. □
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to warmly thank all the reviewers for their insightful and constructive comments.
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