Final-value problem for a weakly-coupled system of structurally damped waves

Abstract

We consider the final-value problem of a system of strongly-damped wave equations. First of all, we find a solution of the system, then by an example we show the problem is ill-posed. Next, by using a filter method, we propose stable approximate (regularized) solutions. The existence, uniqueness of the corresponding regularized solutions are obtained. Furthermore, we show that the corresponding regularized solutions converge to the exact solutions in L² uniformly with respect to the space coordinate under some a priori assumptions on the solutions.