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.


Introduction
Our main interest lies in the following nonlocal porous medium equation with inner absorption term: u t Δu m u p Ω u q y, t dy − ku r , x, t ∈ Ω × 0, ∞ , 1.1 subjected to weighted linear nonlocal boundary and initial conditions, where m > 1, p ≥ 0, q > 0, p q ≥ 1, r ≥ 1, k > 0, and Ω ⊂ R N N ≥ 1 is a bounded domain with smooth boundary. The weight function f x, y / ≡ 0 is a nonnegative continuous 2 Abstract and Applied Analysis function defined on ∂Ω × Ω, and Ω f x, y dy > 0 on ∂Ω. The initial value u 0 x ∈ C 2 α Ω with 0 < α < 1 is a nonnegative continuous function satisfying the compatibility condition on ∂Ω.
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, u x, t , Δu m , u p Ω u q y, t dy, and −k represent the density of the species, the mutation, the human-controlled distribution, and the decrement rate of biological species at location x and time t, 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 u x, t 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 8 studied the linear parabolic subjected to the nonlocal Dirichlet boundary condition 1.2 , where A is an elliptic operator, He proved that when Ω f x, y dy ≤ ρ < 1, the solution tends to 0 monotonously and exponentially as t → ∞. With regard to more general discussions on initial boundary value problem for linear parabolic equation with nonlocal Neumann boundary condition, one can see 9 by Pao where the following problem was considered: 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 11 considered the semilinear parabolic equation x, t ∈ Ω × 0, ∞ , 1.9 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 g u under suitable assumption; Cui and Yang 12 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 13 and the references therein.
Recently, Wang et al. 14 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 q ≥ 1, r ≥ 1, and α > 0. By using comparison principle and the method of upper-lower solutions, they got the following results.
a If 1 ≤ q < r, then the solution of the problem exists globally.
b If q > r ≥ 1, the problem has solutions blowing up in finite time as well as global solutions. That is, i if Ω f x, y dy ≤ 1, and u 0 x ≤ α/|Ω| 1/ r−q , then the solution exists globally; ii if Ω f x, y dy > 1, and u 0 x > α/ |Ω| − α 1/r |Ω| > α , then the solution blows up in finite time; iii for any f x, y ≥ 0, there exists a 2 > 0 such that the solution blows up in finite time provided that u 0 x > a 2 φ x , where φ x is the corresponding normalized eigenfunction of −Δ with homogeneous Dirichlet boundary condition, and Ω φ x dx 1.
c If q r > 1.
i The solution blows up in finite time for any f x, y ≥ 0 and large enough u 0 .
ii If Ω f x, y dy < 1, the solution exists globally for u 0 x ≤ a 1 Φ x for some a 1 > 0, where Φ x solves the following problem: In addition, for the initial boundary value problem of 1.11 with weighted nonlinear boundary condition and Dirichlet boundary condition, we refer to 15, 16 and references therein, respectively.
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. 3 If m < p q, 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 u 0 x : Abstract and Applied Analysis 5 C 2 there exists a constant δ > 0, such that where δ will be determined later.
Theorem 1.6. Suppose that p q > max{m, r}, Ω f x, y dy ≤ 1 for x ∈ ∂Ω, and the initial data satisfies the conditions C 1 -C 2 , then 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 blowup 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.

Comparison Principle and Local Existence
In this section, we establish a suitable comparison principle for problem 1. The following comparison principle plays a crucial role in our proofs which can be obtained by establishing suitable test function and Gronwall's inequality.

2.7
and ξ is a function between Ω f x, y udy and Ω f x, y udy. Noticing that u x, t and u x, t are bounded functions, it follows from m > 1, p ≥ 1, q ≥ 1, and r ≥ 1 that Φ i i 1, 2, 3, 4 are bounded nonnegative functions. If 0 ≤ p < 1 or 0 < q < 1, we have Φ 2 ≤ δ p−1 and Φ 4 ≤ δ q−1 by the condition that u x, 0 ≥ 0 or u x, 0 ≥ δ > 0. Thus, we may choose an appropriate ψ x, t as in 17, p. 118-123 to obtain Abstract and Applied Analysis   7 where ω max{ω, 0} and C 1 , C 2 > 0. It follows from u x, 0 ≤ u x, 0 that By Gronwall's inequality, we know that 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 u 0 x ∈ C 2 α Ω ∩ C Ω 0 < α < 1 satisfies the compatibility condition. Then, there exists a constant T * > 0 such that the problem

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 p q > r, we know that v p q 1 > v r , and |Ω|v p q − kv r ≥ |Ω| − k v p q − k. Therefore, the solution of 3.1 is an upper solution of the following problem: When |Ω| > k and p q > 1, it is known that the solution to the problem 3.2 blows up in finite time It is obvious that the solution of problem 3.1 is a lower solution of problem 1.1 -1.3 when Ω f x, y dy ≥ 1 and u 0 x > v 0 . By Proposition 2.2, u x, t is a blow-up solution of problem 1.1 -1.3 .
Proof of Theorem 1.3. 1 The case of p q > r. Let u k/|Ω| 1/ p q−r . It is easy to show that if Ω f x, y dy < 1 and u 0 x < k/|Ω| 1/ p q−r , u x, t is the upper solution of problem 1.1 -1.3 , then we can draw the conclusion.
2 The case of p q ≥ max{m, r}. We need to establish a self-similar blow-up solution in order to prove the blow-up result. We first suppose that ω ∈ C 1 Ω , ω x ≥ 0, and ω x is not identically zero, and ω x | ∂Ω 0. Without loss of generality, we assume that 0 ∈ Ω and ω 0 > 0. Let for sufficiently small T > 0 and R A A 2 . Calculating the derivative of u, we obtain where M B 0,R V q/m |ξ| dξ > 0. It is easy to see that V ξ ≤ 0 and V < 1, and

3.5
Abstract and Applied Analysis 9 Since p q ≥ m > 1, choosing γ > 0 such that m < 1 γ /γ < p q, and μ is sufficiently small such that Then, for small enough T > 0, we have If x ∈ ∂Ω, ω 0 > 0, and ω is continuous, it is known that there exist positive ε and ρ such that ω ≥ ε for x ∈ B 0, ρ . We can get B 0, RT σ ⊂ B 0, ρ ⊂ Ω if T is small enough. Then, u ≤ Ω f x, y udy on ∂Ω × 0, T . It follows from 3.3 that u x, 0 ≤ K 0 ω x for sufficiently large K 0 . Therefore, one can observe that the solution to 1.1 -1.3 exists no later than t T provided that u 0 ≥ K 0 ω x . This implies that the solution blows up in finite time for large enough initial data.
Proof of Theorem 1.4. Suppose that λ 1 > 0 is the first eigenvalue of −Δ with homogeneous Dirichlet boundary condition, and φ x is the corresponding eigenfunction. Let M Ω 1/ φ y ε dy ≤ 1 for some 0 < ε < 1, where M max x∈∂Ω,y∈Ω f x, y . Now, we assume that u m v, then 1.
Set v x, t Ce γt/n / φ x ε , where C is determined later, then it follows that v n t C n γe γt φ x ε n , 3.8

Abstract and Applied Analysis
And, we can get φ ε nq dy kC nr e γrt φ ε nr .

3.14
It is obvious that the global existence result holds for k ≥ |Ω|.

3.19
Through the previous discussion, we know that the global existence results hold.

Abstract and Applied Analysis
For the blow-up case of m < p q, it holds clearly from the second part of the proof of Theorem 1.3.

Blow-Up Rate Estimates
Next, we will get the following precise blow-up rate estimates for slow diffusion case under some suitable conditions.
Let v u m , we just need to consider the problem * - * * * , and let pn p 1 , qn q 1 , and rn r 1 , then * becomes v n t Δv v p 1 Ω v q 1 y, t dy − kv r 1 , x ∈ Ω, t > 0. 4.1 Suppose that v x, t is the blow-up solution of problem * - * * * in finite time T , and set V t max x∈Ω v x, t .
Proof of Theorem 1.6. 1 We can easily know that V t is Lip continuous and differential almost everywhere, Then, it follows that V t ≤ 1 n |Ω|V p 1 q 1 1−n .