Blow-up Phenomena For Pseudo-Parabolic Kirchhoff Equations With Logarithmic Nonlinearity

Abstract

We consider pseudo-parabolic equations with $r(x)$-Kirchhoff coefficient and
logarithmic nonlinear term subject to Dirichlet boundary condions%
\begin{equation*}
\upsilon _{t}-\mu \Delta \upsilon _{t}-M(\left( \left\Vert \nabla \upsilon
\right\Vert _{r(x)}^{r(x)}\right) \Delta _{r(x)}\upsilon =\left\vert
\upsilon \right\vert ^{s(x)-2}\upsilon \ln \left\vert \upsilon \right\vert .
\end{equation*}%
Using a method based on differential inequalities, we prove that the
solutions become unbounded at a finite time $T$, and, we ascertain an upper
limit for this time in the case of negative initial energy. Additionally, we
determine a lower limit for the time at which blow-up occurs.