Energy method in the partial Fourier space and application to stability problems in the half space (Q619883)

The authors study the asymptotic stability of planar stationary waves, for the half space problem of the damped wave equation with a nonlinear convection term \(\partial_t^2 u-\Delta u+\partial_t u+\nabla\cdot f(u)=0\), with boundary condition \(u(0,x',t)=u_b\), and initial conditions \(u(x,0)=u_0(x)\), \(\partial_t u(x,0)=u_1(x)\). Here \(x=(x_1,x')\in \mathbb{R}_+\times\mathbb{R}^{n-1}\) and \(n\geq 2\), \(u_b\) is a constant and \(u_0(x)\rightarrow u_+\neq u_b\) as \(x_1\rightarrow \infty\). The planar stationary wave for this problem is the one dimensional stationary solution \(u(x,t)=\phi(x_1)\). The idea is to prove a sharp decay estimate for the perturbation \(v(x,t)=u(x,t)-\phi(x_1)\). For this purpose, the authors develop the energy method (with multipliers \(v_t\) and \(v\)) in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable \(x'\). For the variable \(x_1\), the authors use \(L^2\) or weighted \(L^2\) spaces.