Nonexistence of global solutions for an inhomogeneous pseudo-parabolic equation

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $u_t-kΔu_t-Δu=I^γ_{0+}(|u|^{p})+ω(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$ where $p>1,\,k\geq 0$, $ω(x)\neq0$ and $I^γ_{0+}$ is the left Riemann-Liouville fractional integral of order $γ\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $γ=0,\,p=p_c$ 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 $γ>0$, the problem does not admit global solutions for any $p>1$ and $\int_{\mathbb{R}^N}ω(x) dx>0.$

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