Convergence Rates And Stability Of Boundary Layer Solution For Damped Wave Equation In Half Space

This paper is concerned with the initial-boundary value problem for the damped wave equations with a nonlinear convection term in the half space R+In this paper,u(t,x)is an unknown function of t∈(0,+∞)and x∈R+,v+≠ub are two given constant states and the nonlinear function f(u)is a general smooth function which may have more than one inflection points.For the non-degenerate case f’(u+)<0, it is shown in Ref.[1]that the above initial-boundary value problem admits a unique global solution u(t,x)which converges to the stationary wave Φ(x)uniformly in x∈R+as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave δ=|u+-ub|are sufficiently small and the strict convexity of f(u).Therefore,in order to make this damped wave equation demonstrate its advantages and role in image denoising and physical applications,we consider a weaker condition. We do not ask for a sufficiently small δ and the strict convexity of f(u). Firstly,we use basic energy methods and continuity techniques to prove this initial-boundary value prob-lem,which exists a unique global smooth solution u(t,x). This u(t,x)exists a initial-boundary value solution which tends to Φ(x)when t→+∞. Secondly,we use space-weighted energy method to make a further assume that,for any constant α>0,we have (1+x)α/2(?)i/(?)xi(u0(x)-Φ(x))∈L2(R+)(i=0,1),(1+x)α/2(?)i/(?)xiu1∈L2(R+)(i=0,1). Then with t→+∞, we have algebraic convergence‖u(t,x)-Φ(x)‖H2(R+)≤CEα(1+t)α/2and exponential convergence‖u(t,x)-Φ(x)‖H2(R+)≤CEα.expe-β/2t We can see that we also obtain the corresponding nonlinear stability and decay estimate. Our analysis is based on the space-time weighted energy method and some delicate energy estimates.