On the Density of the Range of the Semigroup for Semilinear Heat Equations

Abstract

We consider the semilinear heat equation u t − Δu + f(u) = 0 in a bounded domain Ω ⊂ R n , n ≥ 1, for t > 0 with Dirichlet boundary conditions u = 0 on ∂Ω × (0,∞). For T > 0 fixed we consider the map S(T) : C 0 (Ω) → C 0 (Ω) such that S(T )u 0 = u (x, T) where u is the solution of this heat equation with initial data u(x, 0) = u 0(x) and C 0(Ω) is the space of uniformly continuous functions on Ω that vanish on its boundary. When f is globally Lipschitz and for any T > 0 we prove that the range of S(T)is dense in C 0(Ω). Our method of proof combines backward uniqueness results, a variational approach to the problem of the density of the range of the semigroup for linear heat equations with potentials and a fixed point technique. These methods are similar to those developed by the authors in an earlier paper in the study of the approximate controllability of semilinear heat equations.