New General Decay Result for a Class of Viscoelastic Pseudo-parabolic Equations with Critical Sobolev Exponent

This work is concerned with a multi-dimensional viscoeastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literatures. More precisely, we consider the kernel $g:\left[ {0,\infty } \right) \longrightarrow \left( {0,\infty } \right)$ satisfying ${g^\prime }\left( t \right) \leqslant - \xi \left( t \right)G\left( {g\left( t \right)} \right)$, where $\xi$ and $G$ are functions satisfying some specific properties.