NONLINEAR STABILITY OF PLANAR BOUNDARY LAYER SOLUTIONS FOR DAMPED WAVE EQUATION

Abstract

We study the global stability of planar boundary layer solutions to the initial boundary value problem for the damped wave equation, in presence of a nonlinear convection term in the two-dimensional half space R+× R. By employing the energy method and a continuation argument, we establish that such an initial boundary value problem admits a unique global solution for a class of large initial perturbations, and that this global solution converges to the corresponding planar boundary layer solution, uniformly in (x, y) ∈ R+× R as t tends to infinity — provided the strength of the planar boundary layer solution is suitably small. Moreover, by exploiting the space–time weighted energy method and the properties of the planar boundary layer solutions, we also derive a convergence rate (both algebraic and exponential in nature) for the non-degenerate case.