The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time- dependent dissipation term of the form u"+Au " +Au +b(t)u' ' = g(t) in a real Hilbert space His considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).