A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation ut=Δu+uput=Δu+up on RNRN, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈RNx∈RN and t∈Rt∈R. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t⩾0t⩾0, then it necessarily converges to 0, as t→∞t→∞, uniformly with respect to x∈RNx∈RN.