Nonlinear dissipative wave equations with potential

We study the long time behavior of solutions of the wave equation with absorbtion |u|p−1u and variable damping V (x)ut, where V (x) ∼ V0|x|−α for large |x|, V0 > 0. For α ∈ [0, 1) and 1 < p < (n + 2)/(n − 2) we establish decay estimates of the energy, L2 and Lp+1 norm of solutions. We find three different regimes of decay of solutions depending on the exponent of the absorbtion term. We show that p1(n, α) := 1 + 2(4 − α2)/[(n − α)(4 − α)] is a critical exponent in the following sense. For the supercritical region, namely p1(n, α) < p < (n+2)/(n−2), the decay of solutions of the nonlinear equation coincides with the decay of the corresponding linear problem. For the subcritical region 1 < p < p1(n, α) the decay is much faster. Moreover, the subcritical region 1 < p < p1(n, α) is divided by two subregions with completely different decay rates by another critical exponent p2(n, α) := 1+2α/(n−α). The decay rate of solutions becomes independent of α if 1 < p < p2(n, α). Deriving the decay of solutions even for the linear problem with potential V (x) is a delicate task and requires serious strengthening of the multiplier method. We also consider the decay problems for nonlinear damped equation with absorbtion under more general assumptions on the potential.