Critical exponents for a semilinear parabolic equation with variable reaction

Abstract

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:

where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.

When Ω = ℝ^N we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.

When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).