Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

Abstract

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t  = div(|∇u ^m | ^p−2∇u ^m ) + a∫_Ω u ^q (y, t)dy with a, q, m > 0, p > 1, m(p − 1) < 1, in a bounded domain Ω ⊂ R ^N (N > 2). More precisely speaking, it is shown that if q > m(p − 1), any non-negative solution with small initial data vanishes in finite time, and if 0 < q < m(p − 1), there exists a solution which is positive in Ω for all t > 0. For the critical case q = m(p − 1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ = ∫ _Ωφ ^p−1(x)dx and φ is the unique positive solution of the elliptic problem −div(|∇φ| ^p−2∇φ) = 1, x ∈ Ω; φ(x) = 0, x ∈ ∂Ω.

Keywords:

• polytropic filtration equation
• nonlocal source
• extinction

AMS Subject Classifications::

• 35B33
• 35K55

Acknowledgements

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. The first author is also supported by Graduate Innovation Fund of Jilin University (20111034).