General decay for a nonlinear pseudo-parabolic equation with viscoelastic term

Abstract

This work is concerned with a multi-dimensional viscoelastic 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 literature. More precisely, we consider the kernel g:[0,∞)⟶(0,∞)g:{[}0,\infty )\hspace{0.33em}\longrightarrow \hspace{0.33em}(0,\infty ) satisfying g′(t)⩽−ξ(t)G(g(t)){g}^{^{\prime} }(t)\leqslant -\xi (t)G(g(t)), where ξ\xi and GG are functions satisfying some specific properties.

Keywords

• pseudo-parabolic equation
• viscoelastic
• global existence
• decay rate
• 35b40
• 35k51