Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition
Boundary Value Problemsvolume 2014, Article number: 8 (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.
In the past decades, there have been many works dealing with global existence and blow-up properties of solutions for nonlinear parabolic equations, especially the initial boundary value problems with nonlocal terms in equations or boundary conditions, we refer to [1–10] and references therein. For the study of an initial boundary value problem for local parabolic equations with nonlocal boundary condition, we refer to [1–4]. For example, Friedman  studied the linear parabolic equation
subject to the weighted linear nonlocal Dirichlet boundary condition
where A is an elliptic operator,
and the nonnegative continuous function satisfies suitable conditions. He proved that when , the solution approaches to 0 monotonously and exponentially as . As regards more general discussions on an initial boundary value problem for a linear parabolic equation with a weighted linear nonlocal Neumann boundary condition, one can refer to  by Pao, where the following problem:
was considered. He studied the asymptotic behavior of solutions and found the influence of the weight function on the existence of global and blow-up solutions. Later, Akila  adopted the method of an upper-lower solution to consider the semilinear parabolic equation
under a similar weighted linear nonlocal boundary condition. Wang et al.  studied a porous medium equation with power form source term,
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 study of initial boundary value problem with nonlocal parabolic equation, especially the nonlocal problem with time-integral, we refer to [5–10]. Under a homogeneous Dirichlet boundary condition, Li and Xie  studied the nonlinear diffusion equation
where , . They obtained the sufficient conditions of global existence and blow-up of solutions under appropriate critical conditions. Furthermore, under the following assumptions:
they derived the following blow-up rate:
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).
Zhou et al.  considered the following singular diffusion equation with memory term:
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].
Recently, Liu and Mu  considered the following semilinear parabolic equation with memory term:
subject to a weighted nonlinear nonlocal boundary,
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.
In view of the works mentioned above, a nonlocal parabolic equation with time-integral term does not seem to be so much investigated as nonlocal equations with space-integral terms. Already at first glance, the problem with a memory term has some difficulties in proving the existence of non-global solutions. First if t is sufficiently small, the nonlinear memory term vanishes, and then it is not clear whether the comparison principle holds in proving the existence of global small solutions. As far as we know, there are a few papers about the blow-up phenomenon for the p-Laplacian evolution equation with nonlinear memory term. Motivated by it, we consider the global existence and blow-up properties of the following p-Laplacian evolution equation with nonlinear memory term and inner absorption term:
subject to weighted linear nonlocal boundary and initial conditions
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.
2 Comparison principle and local 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).
Definition 1 If the nonnegative function satisfies the following conditions:
where is nonnegative, , and .
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 .
Proof For , since and are the lower and upper solutions of problem (1.5)-(1.7), respectively, it follows that
Choose a test function for , where is a characteristic function defined on , then we have
By Lemma 4.10 in , we know for . Moreover, it follows from , , that () is bounded, and if , or , we have , since , . Furthermore, because and are bounded functions, we can get
Since , it follows that
By Gronwall’s inequality, we can deduce that , and so in .
For , , we have
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)
Suppose that , , , and , the nonnegative initial data satisfies the compatibility conditions for and for . Then there exists a constant such that problem (1.5)-(1.7) admits a nonnegative solution for each . Furthermore, either or
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 , .
3 Blow-up solutions and blow-up rate
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.
Proof In order to prove the blow-up result, we need to establish a self-similar blow-up solution. Let
where , , and
It is obvious that blows up at as t approaches T. Set
An explicit calculation yields
Then there exists satisfying
Therefore we have
In view of the above, this gives
We will discuss the problem for two cases.
Case 1. , . We need to show that for sufficiently small T,
Let δ be sufficiently large, satisfying . By the condition , , and , we just have to make the following equality hold:
It is obvious that it holds for .
For , choose , such that
Case 2. , choose and we can get
; we have , then
Since , choose and to satisfy . However,
thus we find
in the sense of and .
In order to get the result, we have to show that
Note that and , we choose
in which case we can get the result.
; we have and . Since and , we know that , and(3.2)
However, since and , it follows that
Finally, we need to show that
Because and , we have to show
In other words, we just need the following inequality:
to hold. So choose T to be small enough, and we can get
For , ,
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.
Proof Consider the following problem:
As , we know , and . Therefore, the solution of equation (3.3) is an upper solution of the following problem:
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 .
Theorem 3 Suppose that for . If , satisfies condition (H1)-(H2), and is the blow-up solution of problem (1.5)-(1.7) in finite time T with , then the blow-up rate is
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 .
Proof It is obvious that is Lipschitz continuous and differentiable almost everywhere. By equation (1.5) with and , it yields
Integrating the result above over , we can obtain the conclusion. □
Proof of Theorem 3 Let , where is sufficiently small. Since , we have
so we can choose to be small enough and thus obtain
On the other hand, as , we get
Let , By Jensen’s inequality, this gives
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. □
4 Global existence of solutions
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.
Proof Let , where are determined later, and solves the following problem:
where . Let , ,
since , choosing
we can infer that
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.
Proof Suppose that solves the following problem:
in which . Let .
Set , where and , then
For , we choose such that and . It follows that
Since , choosing
we can get
On the other hand, for and sufficiently large A, we have
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. □
Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Q. Appl. Math. 1986, 44(3):401-407.
Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum, New York; 1992.
Akila Y: On a nonlocal parabolic problem. Demonstr. Math. 2009, 42(4):745-755.
Wang YL, Mu CL, Xiang ZY: Blow-up of solutions to a porous medium equation with nonlocal boundary condition. Appl. Math. Comput. 2007, 192: 579-585. 10.1016/j.amc.2007.03.036
Li YX, Xie CH: Blow-up for semi-linear parabolic equations with nonlinear memory. Z. Angew. Math. Phys. 2004, 55: 15-27. 10.1007/s00033-003-1128-6
Souplet P: Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory. Z. Angew. Math. Phys. 2004, 55: 28-31. 10.1007/s00033-003-1158-0
Zhou J, Mu CL, Lu F: Blow-up and global existence to a degenerate reaction-diffusion equation with nonlinear memory. J. Math. Anal. Appl. 2007, 333: 1138-1152. 10.1016/j.jmaa.2006.12.007
Du LL, Mu CL: Global existence and blow-up analysis to a degenerate reaction-diffusion system with nonlinear memory. Nonlinear Anal., Real World Appl. 2008, 9: 303-315. 10.1016/j.nonrwa.2006.10.005
Zhou J, Mu CL, Fan MS: Global existence and blow-up to a degenerate reaction-diffusion system with nonlinear memory. Nonlinear Anal., Real World Appl. 2008, 9: 1518-1534. 10.1016/j.nonrwa.2007.03.016
Liu DM, Mu CL: Blow-up analysis for a semi-linear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition. Electron. J. Qual. Theory Differ. Equ. 2010, 51: 1-17.
Bebernes J, Eberly D: Mathematical Problems from Combustion Theory. Springer, New York; 1989.
Furter J, Grinfield M: Local vs. non-local interactions in populations dynamics. J. Math. Biol. 1989, 27: 65-80. 10.1007/BF00276081
Calsina A, Perello C, Saldana J: Non-local reaction-diffusion equations modelling predator-prey co-evolution. Publ. Mat. 1994, 32: 315-325.
Allegretto W, Fragnelli G, Nistri P, Papin D: Coexistence and optimal control problems for a degenerate predator-prey model. J. Math. Anal. Appl. 2011, 378: 528-540. 10.1016/j.jmaa.2010.12.036
Kastenberg WE: Space dependent reactor kinetics with positive feed-back. Nukleonika 1968, 11: 126-130.
Pao CV: Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comput. Appl. Math. 1998, 88(1):225-238. 10.1016/S0377-0427(97)00215-X
Pao CV: Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comput. Appl. Math. 2001, 136(1-2):227-243. 10.1016/S0377-0427(00)00614-2
Diaz JI: Nonlinear Partial Differential Equations and Free Boundaries: Elliptic Equations. Vol. I. Pitman, London; 1985.
Soupllet P: Blow-up in nonlocal reaction-diffusion equations. SIAM J. Math. Anal. 1998, 29(6):1301-1334. 10.1137/S0036141097318900
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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.