Critical extinction exponents for a fast diffusion equation with nonlocal source and absorption

Haixia Li,Yuzhu Han

doi:10.1186/1687-2770-2014-24

Abstract

In this article, the authors apply the super-solution and sub-solution methods, instead of energy estimate methods, to investigate the critical extinction exponents for a fast diffusion equation with a nonlocal source and an absorption term. They give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve, in some sense, the results by Xu et al. (Bound. Value Probl. 2013:24 2013) and by Han et al. (Arch. Math. 97:353-363, 2011).

Highlights

1 Introduction In this paper, we investigate the following fast diffusion equation with a nonlocal source and an absorption term:
The equation in ( . ) is a fast diffusion equation perturbed by both a nonlocal source term and an absorption term, which describes the diffusion of concentration of some Newtonian fluids or the density of some biological species
What we are interested in here is the extinction in finite time of the nonnegative solutions u(x, t) of ( . ), i.e. there exists a finite time T > such that the solution is nontrivial for < t < T, but u(x, t) ≡ for almost every (x, t) ∈ × [T, ∞)

Summary

Introduction

) vanishes in finite time for any nonnegative initial data provided that either | | or a is sufficiently small; If < r < , the nonnegative nontrivial weak solution of Problem ) vanishes in finite time provided that u , | | or a is sufficiently small, and q > C with C being a positive constant depending only on N , r and m; If < r < and m > q ≥ r, the nonnegative nontrivial weak solution of Problem If q > m, all the solutions u(x, t) vanish in finite time for suitably small initial data u (x).

Results

Conclusion