\(L^{2}\)-estimates of solutions for damped wave equations with space-time dependent damping term (Q1047685)

The Cauchy problem to the equation \(u_{tt}-\Delta u +b_0(1+\left|x\right|^2)^{-\alpha /2}(1+t)^{-\beta}u_t+\left|u\right|^{\rho -1}u=0\) in \(\mathbb R^{+}\times\mathbb R^n\), where \(\alpha ,\beta \geq 0,\, \alpha+\beta <1\), is considered. Using the local existence theorem and the weighted energy method the authors prove global existence and \(L^2\)-decay rate of the solution. Throughout the paper the exponent \(\rho \) yields: \(1<\rho <\infty \) for \(n=1,2\) and \(1<\rho \leq \frac{n}{n-2}\) for \(n\geq 3\).