Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation u_t = Δu + |u| ^p-1u. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.