Nonlinear Stability And Decay Rates Toward Planar Boundary Layer Solutions For Damped Wave Equation With Large Initial Perturbation

This paper is concerned with global stability of planar boundary layer solutions to the initial-boundary value problem for the damped wave equation with a nonlinear convection term in two-dimensional half space R+×RWe first show that the above the initial-boundary value problem has a unique local solution by employing the standard contraction-mapping principle, then by employing the basic energy method and a continuation argument, we can show that the above initial-boundary value problem admits a unique global solution u(t,x,y) for a class of large initial perturbation and such a u(t, x, y) converges to the corresponding planar boundary layer solutionφ(x) uniformly in (x, y)∈R+×R as t tends to infinity provided that the strength of the planar boundary layer solution is suitably small. Hereφ(x) is the unique solution of the following problemMoreover, by exploiting the space-time weighted energy method and the proper-ties of the planar boundary layer solutions, the convergence rate (both algebraic and exponential) of u(t,x,y) towardφ(x) are also obtained for the non-degenerate case f'1(u+)<0.