L²-estimates of solutions for damped wave equations with space-time dependent damping term

In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |^{ρ - 1} u. Here, b (t, x) = b₀ (1 + | x |²)^{- frac(α, 2)} (1 + t)^{- β} with b₀ > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L² decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.