Limits of solutions to the semilinear wave equation with small parameter

We study the existence of the limits of solution to singularly perturbed initial boundary value problem of hyperbolic – parabolic type with boundary Dirichlet condition for the semilinear wave equation. We prove the convergence of solutions and also the convergence of gradients of solutions to perturbed problem to the corresponding solutions to the unperturbed problem as the small parameter tends to zero. We show that the derivatives of solution relative to time-variable possess the boundary layer function of the exponential type in the neighborhood of $t=0$.