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

We consider the semilinear parabolic equation ut = ∆u+ up on RN , where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x ∈ RN and t ∈ 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 ≥ 0, then it necessarily converges to 0, as t →∞, uniformly with respect to x ∈ RN .