WELL-POSEDNESS AND BLOW-UP FOR AN IN-HOMOGENEOUS SEMILINEAR PARABOLIC EQUATION

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation u(i) - Delta u = vertical bar x vertical bar(alpha)vertical bar u vertical bar(p) + sigma(t)w(x) in (0, infinity) x R-N, where N >= 3, p > 1, alpha > -2, sigma,w are continuous functions such that sigma(t) = t(sigma) or sigma(t) similar to t(sigma) as t -> 0, sigma(t) similar to t(m) as t -> infinity. We obtain local existence for sigma > -1. We also show the following: If m <= 0, p < and N-2m+alpha/N-2m-2 and integral(RN) w(x) dx > 0, then all solutions blow up in finite time; If m > 0, p > 1 and integral(RN) w(x) dx > 0, then all solutions blow up in finite time; If sigma(t) = t(sigma) with -1 < sigma < 0, then for u(0) := u(t = 0) and w sufficiently small the solution exists globally. We also discuss lower dimensions. The main novelty in this paper is that blow up depends on the behavior of at infinity.