Fast energy decay for wave equations with variable damping coefficients in the 1-D half line. (Q265205)

The paper deals with the asymptotic behavior of the solution to the initial boundary value problem to the equation NEWLINE\[NEWLINEu_{tt}-u_{xx}+V(x)u_t=0,\quad (t,x)\in (0,\infty)\times (0,\infty)NEWLINE\]NEWLINE satisfying the Dirichlet boundary condition and compactly supported initial data. Let there exist constants \(V_0>0,V_1\in [V_0,\infty), L_2>0\) so that \(V(x)\) satisfies the inequalities NEWLINE\[NEWLINE\frac{V_0}{1+x}\leq V(x)\leq \frac{V_1}{1+x},\quad x\in [L_2,\infty).NEWLINE\]NEWLINE Using very clever estimates the authors prove the two theorems: If \(V_0>2\) then \(E(t)=O(t^{-2})\) and if \(0<V_0\leq 2\) then \(E(t)=O(t^{-V_0+\delta})\) for a small \(\delta >0\), where \(2E(t):=\| u_t(t,.)\| ^2+\| u_x(t,.)\| ^2\).