Smoothness effect and decay on a class of nonlinear evolution equation
Solutions of the nonlinear evolution equation \(u_{tt}+ M(\| A^{1/2} u(t)\|^ 2)Au+ Au_ t=0\), \(u(0)=u_ 0\), \(u_ t(0)=u_ 1\), are considered, where \(A\) is a positive selfadjoint operator on a Hilbert space. First the author proves existence and uniqueness of global solutions. The main result of the paper is concerned with the smoothing property of the solutions, i.e. under suitable conditions for \(M\) the solutions are as smooth as one can at best expect, namely \(u(t)\) is the intersection of the domains of all powers of \(A\) for all \(t>0\), no matter how smooth the initial data are. Moreover it is proven that the solutions decay uniformly as time goes to infinity.
positive selfadjoint operator on a Hilbert space
existence and uniqueness of global solutions
scientific article; zbMATH DE number 202538
scientific article