Decay of Heat Semigroups in L∞ and Applications to Nonlinear Parabolic Problems in Unbounded Domains

We characterize all domains Ω of RN such that the heat semigroup decays in L(L∞(Ω)) or L(L1(Ω)) as t→∞. Namely, we prove that this property is equivalent to the Poincaré inequality, and that it is also equivalent to the solvability of −Δu=f in L∞(Ω) for all f∈L∞(Ω). In particular, under mild regularity assumptions on Ω, these properties are equivalent to the geometric condition that Ω has finite inradius. Next, we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains. First, we consider the quenching problem for singular parabolic equations. We prove that the solution in Ω quenches in finite time no matter how small the nonlinearity is, if and only if Ω does not fulfill the Poincaré inequality. Second, for the semilinear heat equation with a power nonlinearity, we prove, roughly speaking, that the trivial solution is stable in L∞ or in L1 if and only if Ω has finite inradius.