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.

Keywords

• Nonlinear parabolic equations
• nonlinear boundary conditions
• sub and super-solutions
• interpolation inequalities
• a priori estimates
• bounded solutions