Exponential decay and blow-up results for a nonlinear heat equation with a viscoelastic term and Robin conditions
Tom 119 / 2017
Annales Polonici Mathematici 119 (2017), 121-145 MSC: 34B60, 35K55, 35Q72, 80A30. DOI: 10.4064/ap4084-3-2017 Opublikowany online: 14 June 2017
We consider a nonlinear heat equation with a viscoelastic term and Robin conditions. First, we prove existence and uniqueness of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, we give a sufficient condition for the global existence and exponential decay of weak solutions. The main tools are the Faedo–Galerkin method and defining a modified energy functional together with the technique of Lyapunov functional.