GLOBAL EXISTENCE AND LOCAL CONTINUITY OF SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS

We study the semilinear parabolic equation del(A del u) + V f(u) - u(t ) = 0. The potential V = V(x,t) is allowed to be in a new and general functional class characterized by one integrability condition, which c ontrols both the global growth and local singularity of V while allowi ng global existence and local continuity for solutions of the equation . Locally, the class we are proposing contains certain parabolic Morre y-Campanato spaces. Globally, the class is also general enough to allo w V under some growth conditions proposed by several other authors. Ex amples of V include V(x,t) = 1/(1 + \x\(2+epsilon)) and all functions in L-1(R-1) boolean AND L-infinity(R-1) which includes, for example, V (x,t) = 1/(1 + t(alpha)) where epsilon > 0 and alpha > 1. In fact the condition we are proposing is essentially optimal under certain condit ions.