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 $$u_t-k\Delta u_t-\Delta u=I_{0+}^{\gamma }(|u{|}^p)+\omega (x),(t,x)\in (0,\infty )\times {\mathbb{R}}^N,$$ where $p>1,k\ge 0$, $\omega (x)\ne 0$ and $I_{0+}^{\gamma }$ 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 $N\ge 3$, 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 ${\int }_{{\mathbb{R}}^N}\omega (x)dx>0.$

7 pages, comments are welcome