Critical Exponents of Fujita Type for Inhomogeneous Parabolic Equations and Systems

We consider the large-time behavior of sign-changing solutions of inhomogeneous parabolic equations and systems. For example, for ut = u + u p + w x in Rn × 0 T , we show the following: If n ≥ 3 and Rn w x dx > 0 and 1 < p ≤ n/ n − 2 , then all solutions blow up in finite time, while if p > n/ n − 2 there are both global and nonglobal solutions. We show by example that global solutions exist for all p > 1 and w satisfying Rn w x dx < 0. When n = 1 2 and Rn w x dx > 0, no solution can exist for all time. Extensions of the above result to various geometries and some other problems are indicated.