Threshold, subthreshold, and global unbounded solutions of superlinear heat equations

Abstract We consider the semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions, lying on or below the boundary between blow‐up and global existence, respectively. For the Cauchy–Dirichlet problem, we prove the boundedness and decay to zero of any subthreshold solution. This implies, in particular, that all global unbounded solutions—if they exist—are threshold solutions. For the Cauchy problem, these properties fail in general but we show that they become true for a suitably modified notion of threshold. Our results strongly improve known results even in the model case of power nonlinearities, especially in the Sobolev critical and supercritical cases.