Qs. Zhang, GLOBAL EXISTENCE AND LOCAL CONTINUITY OF SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS, Communications in partial differential equations, 22(9-10), 1997, pp. 1529-1557
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.