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
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 (m > 1), fast (0 < m < 1), and linear (m = 1) 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 m = 1 and p = 0, subject to homogeneous boundary condition (3a) or (3b); for the case k = 0, 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 p + q > max {m, s} 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 ℝ3 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 t* if the solution u(x, t) ≥ 0 of (1)–(3a) blows up in finite time t*.
Theorem 1. Let u(x, t) be a classical solution of (1)–(3a) with p + q > max {m, s}; then a lower bound of the blow-up time for any solution which blows up in Ln(p+q−1) norm (n > max {2, (1/(p + q − 1))}) is t* ≥ 1/(2A[η(0)] 2), where A is a suitable positive constant given later and .
Proof. Define an auxiliary function of the form
Taking the derivative of η(t) with respect to t gives
The application of Hölder inequality to the second term on the right hand side of (6) yields
By (7), it follows from (6) that
Let
Now we seek a bound for ∫Ω vn+1dx in terms of η 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]):
Then for some positive constant μ1 to be determined it follows that
Substituting inequality (18) into (17) gives
We next choose μ1 to make the coefficient of ∫Ω vn+δdx vanish and then choose μ2 to make the coefficient of vanish. It follows that
Integrating inequality (21) from 0 to t gives
from which we derive a lower bound for t*:
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 t* if the solution u(x, t) of (1), (2), and (3b) is blow-up in finite time t*.
Theorem 2. Let u(x, t) be a classical solution of (1), (2), and (3b) with p + q > max {m, s}; then a lower bound of the blow-up time for any solution which blows up in Ln(p+q−1) norm is , where K2 and K3 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 (3.3) in [10], we have
Substituting inequality (26) into (25) yields
Applying the following inequality:
Applying inequality (16), we obtain
Recalling (12) and applying inequality (16) again, for a suitable constant μ3, we obtain
By applying (30), it follows from (31) that
Taking
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.
