Influence of weight functions to a nonlocal p-Laplacian evolution equation with inner absorption and nonlocal boundary condition
© Fang and Zhang; licensee Springer 2013
Received: 24 November 2012
Accepted: 30 May 2013
Published: 27 June 2013
In this paper, we investigate an initial boundary value problem for a nonlocal p-Laplacian evolution equation with an inner absorption and weighted linear nonlocal boundary and initial conditions. By using the modified comparison principle and the method of upper-lower solution, we find the influence of weighted function on determining blow-up or not of nonnegative solutions.
where , , , , , and () is a bounded domain with smooth boundary. The weight function in the boundary condition is nonnegative and continuous on , and on ∂ Ω. The initial function is a nonnegative function satisfying the compatibility condition and , where ν denotes the unit outer normal vector on the boundary.
Many natural phenomena, such as the non-Newton flux in the mechanics of fluid, population of biological species, and filtration, have been formulated as nonlocal diffusive equation (1.1); see [1, 2] and references therein. In the diffusion system of some biological species with human-controlled distribution, , , , and −α represent the density of the species at location x and time t, the mutation, the human-controlled distribution, and the decrement rate of biological species, respectively. The evolution of the species at a point of space depends not only on the density of species in a partial region but also in the total region because of the nonlocal boundary condition that arises from the spatial inhomogeneity (cf. [3–5]). 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 entropy per volume of the materia1 (cf. [6, 7]).
By virtue of the upper-lower solution method, they obtained the existence of global solutions, blow-up properties and blow-up rate of the solutions.
and they showed the existence of global solutions, blow-up properties, and blow-up rates of the solutions. For the Dirichlet initial boundary problem of nonlocal evolutionary equation, see [13, 14] and references therein for details.
with weighted linear nonlocal boundary condition (1.2) and initial condition (1.3), where and . By using the comparison principle and the technique of upper-lower solutions, they obtained the following results:
• If , then the global solution of the problem exists.
• If , the problem has solutions that blow up in finite time as well as global solutions, i.e.,
(i) if and , then the global solution exists;
(ii) if and , (), then the solution blows up in finite time;
(iii) for any , there exists such that the solution blows up in finite time provided that , where is the corresponding normalized eigenfunction of −△ with the homogeneous Dirichlet boundary condition and .
• If , then
(i) the solution blows up in finite time for any and large enough ;
Here, is a positive constant such that .
We know from the works mentioned above that the studies of problem (1.1)-(1.3) have not been developed well. The difficulty lies in finding the influence of a weight function in the boundary condition and the competitive relationship between nonlocal source and inner absorption to determine blow-up or not of solutions. Motivated by it, we intend to find the sufficient conditions of the existence of global and blow-up solutions of problem (1.1)-(1.3) by using the method of upper-lower solution, the technique of ordinary differential equation, the method of characteristic function and self-similar lower solution. Here, the key is to establish a modified comparison principle. We will use the suitable test function and Gronwall’s inequality to derive it. Besides, the main contribution of this paper is our results including the nonlocal nonlinearity Hölder (non-Lipschitz) cases for m or , as well as the Lipschitz cases for , and covering all the results of [11, 15]. Our detailed results are as follows.
Theorem 1 Suppose that . If for and (), then the solution of problem (1.1)-(1.3) blows up in finite time.
Theorem 2 Suppose that . If for , then the global solution of problem (1.1)-(1.3) exists for small initial data.
Theorem 3 Suppose that . If for , then the negative global solution of problem (1.1)-(1.3) exists for small initial data, and the nonnegative solution blows up in finite time for arbitrary and sufficiently large initial data.
Theorem 4 Suppose that .
(i) If , , and , then the global solution of problem (1.1)-(1.3) exists for small enough initial data, and if and , then the solution of problem (1.1)-(1.3) blows up in finite time for sufficiently large initial data.
(ii) If , , and , then there exists a global solution of problem (1.1)-(1.3) for small enough initial data, and if and , then the solution of problem (1.1)-(1.3) blows up in finite time for sufficiently large initial data.
Remark 1 It is easy to see that there is no blow-up solution for in Theorem 4.
Remark 2 The exact values of sufficiently large or small initial data stated in the theorems above will be given in the proofs of Section 3.
Remark 3 The function solves problem (3.2)-(3.3) and a is a constant defined in (3.4).
The rest of our paper is organized as follows. In Section 2, with the definitions of weak upper and lower solutions, we give the modified comparison principle for problem (1.1)-(1.3), which is an important tool for our research. The proofs of our main results are given in Section 3.
2 Comparison principle and local existence
Since equation (1.1) is degenerate when , there is no classical solution in general. Hence, it is reasonable to find a weak solution of (1.1). To this end, we first give the following definitions of lower and upper nonnegative weak solutions of problem (1.1)-(1.3). Let , , and let .
Definition 1 If a nonnegative function satisfies the following conditions:
(1) , , , ,
(2) , where ,
(3) , ,
then is called a weak lower solution of problem (1.1)-(1.3).
Similarly, we can define the weak upper solution of problem (1.1)-(1.3). We say that is a weak solution of problem (1.1)-(1.3) in if it is both weak lower and upper solution of problem (1.1)-(1.3) in .
Remark 4 Lemma 1 can be shown in a similar manner as the one used in Lemma 4.10 in .
The following modified comparison principle plays a crucial role in our proofs, which can be obtained by establishing a suitable test function and using Lemma 1 and Gronwall’s inequality.
Proposition 1 (Comparison principle)
Suppose that and are nonnegative weak lower and upper solutions of problem (1.1)-(1.3), respectively, with , , and in . Then we have in .
It follows from Gronwall’s inequality that , and hence, in .
It is obvious that on if in Ω. Therefore, we obtain in . □
Next, we state the local existence and uniqueness theorem without proof.
Theorem (Local existence and uniqueness)
Remark 5 The local existence, in time, of nonnegative solutions of problem (1.1)-(1.3) can be obtained by using the fixed point theorem or the parabolic regular theory to get a suitable estimate in the standard limiting process (see [18, 19]). We can obtain the uniqueness of solution to the problem by the comparison principle stated in Proposition 1. The proof is more or less standard, and so it is omitted here.
3 Proofs of main results
In this section, we give detailed proofs of our main results to problem (1.1)-(1.3) by using the comparison principle and constructing suitable upper and lower solutions.
When and , it is known that the solution of this problem blows up in finite time if .
It is obvious that the solution of problem (3.1) is a lower solution of problem (1.1)-(1.3), when and . Therefore, the solution blows up in finite time by Proposition 1. □
By Proposition 1, we can obtain the result.
This completes the proof. □
Proof of Theorem 3 We first show the existence of a global solution.
(i) Assume that and set . Choosing a constant A such that , one can deduce the desired result by using a similar argument as the one used for (i) of Theorem 1.
(ii) Assume that and set , where is the function defined in (3.2)-(3.4). Choosing a constant A such that and using the same argument as the one used for (ii) of Theorem 1, one can see that there exists a global solution of problem (1.1)-(1.3).
Secondly, we establish a self-similar blow-up solution in order to show a blow-up of the solution. We now assume that , , is not identically zero, and . Without loss of generality, we assume that and .
for sufficiently small .
If , , and ω is continuous, it is known that there exist positive constants ε and ρ such that for . We then have if T is small enough, and so on . It follows from (3.4) that for sufficiently large . Therefore, one can see that the solution of problem (1.1)-(1.3) exists no later than provided that . This implies that the solution blows up in finite time for large enough initial data. □
Proof of Theorem 4 (i) If , choosing A large enough so that , one can see that is the upper solution of problem (1.1)-(1.3), and hence, the result follows.
Choosing such that , one can see that there exists a global solution of problem (1.1)-(1.3).
where , , , , and . It is obvious that , since . Setting , the result follows by the same argument as the one for the case . □
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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