ON A FINAL VALUE PROBLEM FOR A CLASS OF NONLINEAR HYPERBOLIC EQUATIONS WITH DAMPING TERM

This paper deals with the problem of finding the function u(x, t), (x,t) is an element of Omega x [0, T], from the final data u(x,T) = g(x) and u(t) (x, T) = h(x), u(tt) + a Delta(2)u(t) + b Delta(2)u = R(u). This problem is known as the inverse initial problem for the nonlinear hyperbolic equation with damping term and it is ill-posed in the sense of Hadamard. In order to stabilize the solution, we propose the filter regularization method to regularize the solution. We establish appropriate filtering functions in cases where the nonlinear source R. satisfies the global Lipschitz condition and the specific case R(u) = u vertical bar u vertical bar(p-1),p > 1 which satisfies the local Lipschitz condition. In addition, we show that regularized solutions converge to the sought solution under a priori assumptions in Gevrey spaces.