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

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 G G are functions satisfying some specific properties.