Boundedness and stability of global solutions for some superlinear and nonautonomous heat equations

Abstract

We consider the initial-boundary and boundary only problems for some superlinear and nonautonomous heat equations with subcritical nonlinear terms, and extend some of the well known results for autonomous heat equations to the corresponding nonautonomous case. More precisely, we first establish a priori estimates for global positive solutions of the initial-boundary problem for some superlinear and nonautonomous heat equations with subcritical nonlinear terms by introducing two different kinds of rescaling techniques. Next we get the existence of global bounded positive solutions which do not vanish in infinite time for the corresponding boundary only problem, and then we obtain the stability of such global bounded positive solutions by using a technical auxiliary function. This implies that such global bounded solutions are just the threshold solutions lying on the borderline between blow-up and global existence. At last we devote our attention to the more general threshold solutions for the initial-boundary problem of the superlinear and nonautonomous heat equations.