A degenerate parabolic equation with a nonlocal source and an absorption term

Abstract

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx − u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.

Keywords:

• nonlocal source
• absorption term
• blow-up
• global blow-up
• blow-up rate

AMS Subject Classifications::

• 35K20
• 35K55

Acknowledgements

The authors would like to thank the referees and the editor for their valuable suggestions and comments on this article. The project is supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.