Critical exponents for a semilinear parabolic equation with variable reaction

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem: u(t)(x, t) = Delta u(x, t) + (u(x, t))(p(x)) in Omega x (0, T), 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 Omega = 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 Omega is a bounded domain we prove that there are functions p(x) and domains Omega such that all solutions to the problem blow up in finite time. On the other hand, if Omega is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

Critical exponents for a semilinear parabolic equation with variable reaction Articles