Decay estimates for nonlinear wave equations with variable coefficients

Abstract

We study the long-time behavior of solutions to a particular class of nonlinear wave equations that appear in models for waves traveling in a non-homogeneous gas with variable damping. Specifically, decay estimates for the energy of such solutions are established. We find three different regimes of energy decay depending on the exponent of the absorption term |u|p-1u and show the existence of two critical exponents p1 (n, α, β) = 1 + (2 - β) / (n - α) and p2 (n, α) = (n + α) / (n - α). For p > p1 (n, α, β), the decay of solutions of the nonlinear equation coincides with that of the corresponding linear problem. For p1 (n, α, β) > p, the solution decays much faster. The other critical exponent p2 (n, α) further divides this region into two subregions with different decay rates. Deriving the sharp decay of solutions even for the linear problem with potential α(x) is a delicate task and requires serious strengthening of the multiplier method. Here we use a modification of an approach of Todorova and Yordanov to derive the exact decay of the nonlinear equation.