On internal elastic membrane with strong damping

Abstract

In this work we study existence and uniqueness of solutions to the abstract framework related to the Kirchhoff model for small deformations of a membrane \[ u'' +M ( \| u \| ^{2} ) Au+ F(u)+Au'=0, \] when $M$ is degenerate and $A,$ $F$ are operators. Furthermore, we obtain the asymptotic behavior, as $t \rightarrow\infty,$ of solutions in a concrete case.