Nonlinear dissipative wave equations with space-time dependent potential
The paper deals with the initial value problem for the equation \(u_{tt}-\Delta u+a(t,x)u_{t}+| u| ^{p-1}u=0\), \(t>0\), \(x\in \mathbb {R}^n\), where \(a(t,x)=\eta (t)\lambda (x)\), \(\eta (t)\backsim (1+t)^{-\beta }\), \(\lambda (x)\backsim (1+| x| )^{-\alpha }\) and \(\lambda (x)\) is a radial symmetric function. Under some conditions the author proves some decay estimates of the solution of the problem for \(t\rightarrow +\infty \). A decay rate of the \(L^2\)-norm of the solution for \(a(t,x)=b_0(1+| x| ^2)^{-\alpha /2}(1+t)^{-\beta }\) was discussed by \textit{J. Lin, K. Nishihara} and \textit{J. Zhai} in [J. Differ. Equations 248, No. 2, 403--422 (2010; Zbl 1184.35213)].
scientific article; zbMATH DE number 5907672