Nonexistence of global solutions for an inhomogeneous pseudo-parabolic equation

Abstract

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-k\Delta u_t-\Delta u=I^\gamma_{0+}(|u|^{p})+\omega(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $\omega(x)\neq0$ and $I^\gamma_{0+}$ is the left Riemann-Liouville fractional integral of order $\gamma\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $\gamma=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the case $\gamma>0$, the problem does not admit global solutions for any $p>1$ and $\int_{\mathbb{R}^N}\omega(x) dx>0.$