Critical Exponents For A Semilinear Parabolic Equation With Variable Reaction

In this paper, we study the blow-up properties of the nonnegative solutions to the following Cauchy problem:where 0 < p- = (?) p(x)≤p(x)≤(?) p(x) = p_+ is a nonnegative continuous, boundedfunction.In Chapter 3 , we prove the blow-up condition that the solutions blow up in finite time if and only if max{p_+,q} > 1.In Chapter 4 , we consider the Fujita condition: if 1 1 + (?),then there are global solutions for p- > 1 + (?),all solutions blow up in finite time for 0 < p_-≤p_+≤1 + (?),and there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global solutions when p_- < 1 + (?) < p_+.