The Rate at Which the Energy of Solutions for a Class of p-Laplacian Wave Equation Decays

Abstract

We will investigate the decay estimate of the energy of the global solutions to the p-Laplacian wave equation with dissipation of the form $u_{tt}-\text{div}\mathrm{ }({\left|{\nabla }_{x}u\right|}^{p-2}{\nabla }_{x}u)+\sigma (t)(u_t-\text{div}\mathrm{ }({\left|{\nabla }_x{u}_t\right|}^{m-2}{\nabla }_x{u}_t))=0$ under suitable assumptions on the positive function $\sigma$. For this end we use the multiplier method combined with nonlinear integral inequalities given by Martinez; the proof is based on the construction of a special weight function that depends on the behavior of $\sigma$.