Lower Bounds Estimate for the Blow-Up Time of a Slow Diffusion Equation with Nonlocal Source and Inner Absorption

Abstract

We investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. Based on an auxiliary function method and a differential inequality technique, lower bounds for the blow-up time are given if the blow-up occurs in finite time.

1. Introduction

Our main interest lies in the following slow diffusion equation with nonlocal source term and inner absorption term: subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition where is a bounded domain with smooth boundary , is the closure of , , , , , , , is the unit outer normal vector on , and is the possible blow-up time. By the maximum principle, it follows that in the time interval of existence. In the present investigation we derive a lower bound for the blow-up time when for the solutions that blow up.

Equation (1) describes the slow diffusion of concentration of some Newtonian fluids through porous medium or the density of some biological species in many physical phenomena and biological species theories. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1–3]. In the nonlinear diffusion theory, there exist obvious differences among the situations of slow , fast , and linear diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situation, whereas an infinite speed propagation exists in the fast diffusion situation.

The bounds for the blow-up time of the blow-up solutions to nonlinear diffusion equations have been widely studied in recent years. Indeed, most of the works have dealt with the upper bounds for the blow-up time when blow-up occurs. For example, Levine [4] introduced the concavity method, Gao et al. [5] employed the way of combining an auxiliary function method and comparison method with upper-lower solutions method, and Wang et al. [6] used the regularization method and an auxiliary function method. However, the lower bounds for the blow-up time are more difficult in general. Recently, Payne and Schaefer in [7, 8] used a differential inequality technique and an auxiliary function method to obtain a lower bound on blow-up time for solution of the heat equation with local source term under boundary condition (3a) or (3b). Specially, Song [9] considered the lower bounds for the blow-up time of the blow-up solution to the nonlocal problem (1)-(2) when and , subject to homogeneous boundary condition (3a) or (3b); for the case , we refer to [10].

Motivated by the works above, we investigate the lower bounds for the blow-up time of the blow-up solutions to the nonlocal problem (1)-(2) with homogeneous boundary condition (3a) or (3b). Actually, it is well known that if and the initial value is large enough, then the solutions of our problem blow up in a finite time; one can see [11]. Unfortunately, our results are restricted in because of the best constant of a Sobolev type inequality (see [12]).

This paper is organized as follows. In Section 2, we establish problem (1)-(2) with homogeneous Dirichlet boundary condition (3a). Problem (1)-(2) with homogeneous Neumann boundary condition (3b) is considered in Section 3.

2. Blow-Up Time for Dirichlet Boundary Condition

In this section, we derive a lower bound for if the solution of (1)–(3a) blows up in finite time .

Theorem 1. Let be a classical solution of (1)–(3a) with ; then a lower bound of the blow-up time for any solution which blows up in norm is , where is a suitable positive constant given later and .

Proof. Define an auxiliary function of the form with
Taking the derivative of with respect to gives where is the gradient operator.
The application of Hölder inequality to the second term on the right hand side of (6) yields where denotes the volume of .
By (7), it follows from (6) that
Let then and (8) can be written in the from
Now we seek a bound for in terms of and the first and third terms on the right in (11). First, the application of Hölder inequality yields
Using the following Sobolev type inequality (see [12]): with , , and , we obtain
Then for some positive constant to be determined it follows that Next, we use the fundamental inequality to obtain Note the fact that, for some positive constant ,
Substituting inequality (18) into (17) gives Then, by applying inequality (19), it follows from (11) that
We next choose to make the coefficient of vanish and then choose to make the coefficient of vanish. It follows that with
Integrating inequality (21) from 0 to gives
from which we derive a lower bound for :
This completes the proof of Theorem 1.

3. Blow-Up Time for Neumann Boundary Condition

In this final section, we discuss a lower bound for if the solution of (1), (2), and (3b) is blow-up in finite time .

Theorem 2. Let be a classical solution of (1), (2), and (3b) with ; then a lower bound of the blow-up time for any solution which blows up in norm is , where and are suitable positive constants given later, respectively, and .

Proof. We estimate in inequality (14). In a similar way to the process of the derivation of in [10], we have where , , , and is the th component of the unit outer normal vector on . By virtue of Hölder inequality, we get
Substituting inequality (26) into (25) yields
Applying the following inequality: we conclude that
Applying inequality (16), we obtain where and are arbitrary positive constants.
Recalling (12) and applying inequality (16) again, for a suitable constant , we obtain
By applying (30), it follows from (31) that
Taking then combining (32) with (11) gives where We can make and vanish by taking suitable , , and ; then we have Integrating inequality above from 0 to gives from which we derive a lower bound for ; namely, This completes the proof of Theorem 2.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgments

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 deeply thank all the reviewers for their insightful and constructive comments.