Electronic Journal of Differential Equations (Sep 2010)
Bounded solutions of nonlinear parabolic equations
Abstract
We are concerned with bounded solutions existing for all times for nonlinear parabolic equations with possibly nonlinear boundary conditions. A counterexample shows that, without an additional condition, a (weak) maximum principle does not hold for linear problems defined on the entire real line in time. We consider solutions bounded for all times and derive a (weak) maximum principle which is valid on the entire real line. Using comparison techniques, a priori estimates and approximation methods, we prove the existence and, in some cases, positivity and uniqueness of bounded solutions existing for all times.