Continuity of the set of equilibria for non-autonomous damped wave equations with terms concentrating on the boundary

Abstract

In this article we study the behavior of the solutions of non-autonomous damped wave equations when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood collapses toward the boundary as a parameter approaches zero. We prove the continuity of the set of equilibria for these equations. Moreover, if an equilibrium solution of the limit problem is hyperbolic, then we show that the perturbed equation has only one equilibrium solution nearby.

Keywords

• Wave equation
• semilinear elliptic equations
• equilibria
• non-autonomous
• continuity
• concentrating terms