Decay of heat semigroups in L-infinity and applications to nonlinear parabolic problems in unbounded domains

We characterize all domains Omega of R-N such that the heat semigroup decay s in L(L-infinity(Omega)) or L(L-1(Omega)) as t --> infinity. Namely, we pr ove that this property is equivalent to the Poincare inequality, and that i t is also equivalent to the solvability of -Delta u = f in L-infinity(Omega ) for all f is an element of L-infinity(Omega). In particular, under mild r egularity assumptions on Omega, these properties are equivalent to the geom etric condition that Omega has finite inradius. Next, we give applications of this linear result to the study of two nonlinear parabolic problems in u nbounded domains. First, we consider the quenching problem for singular par abolic equations. We prove that the solution in Omega quenches in finite ti me no matter how small the nonlinearity is, if and only if Omega does not f ulfill the Poincare inequality. Second, for the semilinear heat equation wi th a power nonlinearity, we prove, roughly speaking, that the trivial solut ion is stable in L-infinity or in L-1 if and only if Omega has finite inrad ius. (C) 2000 Academic Press.