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Roles of Weight Functions to a Nonlocal Porous Medium Equation with Inner Absorption and Nonlocal Boundary Condition
This work is concerned with an initial boundary value problem for a nonlocal porous medium equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of weight function on whether determining the blowup of nonnegative solutions or not and establish the precise blow-up rate estimates under some suitable condition.
Our main interest lies in the following nonlocal porous medium equation with inner absorption term: subjected to weighted linear nonlocal boundary and initial conditions, where , and is a bounded domain with smooth boundary. The weight function is a nonnegative continuous function defined on , and on . The initial value with is a nonnegative continuous function satisfying the compatibility condition on .
Many natural phenomena have been formulated as nonlocal diffusive equation (1.1), such as the model of non-Newton flux in the mechanics of fluid, the model of population, biological species, and filtration (we refer to [1, 2] and the references therein). For instance, in the diffusion system of some biological species with human-controlled distribution, , and represent the density of the species, the mutation, the human-controlled distribution, and the decrement rate of biological species at location and time , respectively. Due to the effect of spatial inhomogeneity, the arising of nonlocal term denotes that the evolution of the species at a point of space depends not only on the density of species in partial region but also on the total region (we refer to [3–5]). However, 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 material (we refer to [6, 7]).
To motivate our work, let us recall some results of global and blow-up solutions to the initial boundary value problems with nonlocal terms or with nonlocal terms in boundary conditions (we refer to [8–16]). For the study of the initial boundary value problems for the parabolic equations with local terms which subject to the weighted nonlocal linear boundary condition (1.2), one can see [8–10]. For example, Friedman  studied the linear parabolic subjected to the nonlocal Dirichlet boundary condition (1.2), where is an elliptic operator, He proved that when , the solution tends to 0 monotonously and exponentially as . With regard to more general discussions on initial boundary value problem for linear parabolic equation with nonlocal Neumann boundary condition, one can see  by Pao where the following problem was considered: where He studied the asymptotic behavior of solutions and found the influence of weight function on the existence of global and blow-up solutions. Wang et al.  studied porous medium equation with power form source term subjected to nonlocal boundary condition (1.2). By virtue of the method of upper-lower solutions, they obtained global existence, blow-up properties, and blow-up rate of solutions.
For the study of the initial boundary value problems for the parabolic equations with nonlocal terms which subjected to the weighted nonlocal linear boundary condition (1.2), we refer to [11–16]. Lin and Liu  considered the semilinear parabolic equation with nonlocal boundary condition (1.2). They established local existence, global existence, and blow-up properties of solutions. Moreover, they derived the uniform blow-up estimates for special under suitable assumption; Cui and Yang  discussed the nonlocal slow diffusion equation and they built global existence, blow-up properties, and blow-up rate of solutions. For the system of equations, we refer readers to  and the references therein.
Recently, Wang et al.  studied the following semilinear parabolic equation with nonlocal sources and interior absorption term: with weighted linear nonlocal boundary condition (1.2) and initial condition (1.3), where , and . By using comparison principle and the method of upper-lower solutions, they got the following results.(a)If , then the solution of the problem exists globally.(b)If , the problem has solutions blowing up in finite time as well as global solutions. That is,(i)if , and , then the solution exists globally;(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 homogeneous Dirichlet boundary condition, and .(c)If .(i)The solution blows up in finite time for any and large enough .(ii)If , the solution exists globally for for some , where solves the following problem: here is a positive constant such that .
The aim of this paper is to obtain the sufficient condition of global and blow-up solutions to problem (1.1)–(1.3) and to extend the results of the semilinear equation (1.11) to the quasilinear ones. The difficulty lies in finding the roles of weighted function in the boundary condition and the competitive relationship of nonlocal source and inner absorption on whether determining the blowup of solutions or not. Our detailed results are as follows.
Remark 1.2. There may exist a global solution of problem (1.1)–(1.3) for small enough initial data under the condition of Theorem 1.1. Unfortunately, since the weight function satisfies the condition on the boundary, we cannot construct a suitable supersolution of problem (1.1)–(1.3).
Theorem 1.3. Suppose that , if for , then the solution of problem (1.1)–(1.3) exists globally for the initial data . If , then the solution of problem (1.1)–(1.3) blows up in finite time for large enough initial data and arbitrary .
Theorem 1.5. Suppose that for , where is a positive constant and .(1)If , then every nonnegative solution of problem (1.1)–(1.3) exists globally.(2)If , then for , the solution of problem (1.1)–(1.3) exists globally, where are as defined in (3.13).(3)If , the solution of problem (1.1)–(1.3) exists globally for sufficient small initial data while it blows up in finite time for large enough initial data.
In order to show blow-up rate estimate of the blow-up solution, we need the following assumptions on the initial data :(C1) , for ;(C2) there exists a constant , such that where will be determined later.
Theorem 1.6. Suppose that for , and the initial data satisfies the conditions (C1)-(C2), then where .
The rest of our paper is organized as follows. In Section 2, with the definitions of weak upper and lower solutions, we will give the comparison principle of problem (1.1)–(1.3), which is an important tool in our research. The proofs of results of global existence and blow-up of solutions will be given in Section 3. And in Section 4, we will give the blow-up rate estimate of the blow-up solutions.
2. Comparison Principle and Local Existence
In this section, we establish a suitable comparison principle for problem (1.1)–(1.3). Let , and . Firstly, we start with the precise definitions of upper solution and lower solution of problem (1.1)–(1.3).
Similarly, a nonnegative function is an upper solution if it satisfies (2.1)–(2.3) in the reverse order. We say is a solution of problem (1.1)–(1.3) in if it is both an upper solution and a lower solution of problem (1.1)–(1.3) in which is called classical solution.
The following comparison principle plays a crucial role in our proofs which can be obtained by establishing suitable test function and Gronwall’s inequality.
Proof. Let be a nonnegative function with on . Multiplying the inequality in (2.1) by and integrating it on , we get
and similarly, the upper solution satisfies the reversed inequality,
Let , we have
and is a function between and Noticing that and are bounded functions, it follows from , and that are bounded nonnegative functions. If or , we have and by the condition that or . Thus, we may choose an appropriate as in [17, p. 118–123] to obtain
where and . It follows from that
By Gronwall’s inequality, we know that , that is, in .
For , This completes the proof.
Next, we state the local existence and uniqueness theorem without proof.
Theorem 2.3 (local existence and uniqueness). Suppose that the nonnegative initial data satisfies the compatibility condition. Then, there exists a constant such that the problem (1.1)–(1.3) admits nonnegative solution for each . Furthermore, either or
Remark 2.4. The existence of local nonnegative solutions in time to problem (1.1)–(1.3) can be obtained by using the fixed point theorem (see ) or the regular theory to get the suitable estimate in a standard limiting process (see [19, 20]). By the previous comparison principle, we can get the uniqueness of solution to the problem (1.1)–(1.3) in the case of .
3. Global Existence and Blowup of Solutions
Comparing problems with the general homogeneous Dirichlet boundary condition, the existence of weight function on the boundary has a great influence on the global and nonglobal existence of solutions.
Proof of Theorem 1.1. Consider the following problem:
As , we know that , and . Therefore, the solution of (3.1) is an upper solution of the following problem:
When and , it is known that the solution to the problem (3.2) 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 . By Proposition 2.2, is a blow-up solution of problem (1.1)–(1.3).
Proof of Theorem 1.3. (1) The case of . Let . It is easy to show that if and is the upper solution of problem (1.1)–(1.3), then we can draw the conclusion.
(2) The case of . We need to establish a self-similar blow-up solution in order to prove the blow-up result. We first suppose that , and is not identically zero, and . Without loss of generality, we assume that and .
Let , and , where and . We know that for sufficiently small and . Calculating the derivative of , we obtain where .
It is easy to see that and , and Since , choosing such that , and is sufficiently small such that Then, for small enough , we have
If , and is continuous, it is known that there exist positive and such that for . We can get if is small enough. Then, on . It follows from (3.3) that for sufficiently large . Therefore, one can observe that the solution to (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 1.4. Suppose that is the first eigenvalue of with homogeneous Dirichlet boundary condition, and is the corresponding eigenfunction. Let for some , where
Now, we assume that , then (1.1)–(1.3) becomes that where .
Set , where is determined later, then it follows that And, we can get Choosing and the proper such that then
For , so we can obtain that is the upper solution of problem –. From Proposition 2.2, we know that there exists global solution of problem –. Since the problem (1.1)–(1.3) has the same solution with problem –. We know that there exists a global solution of problem (1.1)–(1.3). This completed the proof.
Proof of Theorem 1.5. Suppose that solves the following problem:
where is a positive constant such that . Then, let .
Set , here will be determined later and , It is obvious that the global existence result holds for . For , since , we know that , then (1)If , choosing we have (2)If , selecting , and we get (3)If , we choose such that For , Through the previous discussion, we know that the global existence results hold.
For the blow-up case of , it holds clearly from the second part of the proof of Theorem 1.3.
4. Blow-Up Rate Estimates
Next, we will get the following precise blow-up rate estimates for slow diffusion case under some suitable conditions.
Proof of Theorem 1.6. (1) We can easily know that is Lip continuous and differential almost everywhere,
Then, it follows that
Integrating it over , we get
(2) Next, we set , where , then Since , by Hölder’s inequality, we know that and by Young inequality, we have
Since , we get and , then by (4.7)-(4.8), Choosing , such that
For , because of , we then have Noticing that for and applying Jensen’s inequality to the last part in the previous inequality, we can get Hence, we can get Since satisfies the conditions (C1)-(C2) and , where , then . Combining (4.10)-(4.13), we can know that for , that is, .
Integrating it over , we have then it follows that where . This completed the proof.
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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