Decay Estimates for Nonlinear Wave Equations with Variable Coefficients

We studied the long time behavior of solutions of nonlinear wave equations with variable coefficients and an absorption nonlinearity. Such an equation appears in models for traveling waves in a non-homogeneous gas with damping that changes with position. We established decay estimates of the energy of solutions. We found three different regimes of decay of solutions depending on the exponent of the absorption term. We show the existence of two critical exponents. For the exponents above the larger critical exponent, the decay of solutions of the nonlinear equation coincides with that of the corresponding linear problem. For exponents below the larger critical exponent, the solution decays much faster. Lastly, the subcritical region is further divided into two subregions with different decay rates. Deriving the sharp decay of solutions even for the linear problem with potential is a delicate task and requires serious strengthening of the multiplier method. Here we used a modification of an approach of Todorova and Yordanov to derive the exact decay of the nonlinear equation.