In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $u_t-k\Delta u_t-\Delta u=I_{0+}^{\gamma }\left({\left|u\right|}^p\right)+\omega \left(x\right),(t,x)\in (0,\infty )\times {\mathbb{R}}^N,$where $p>1,k\ge 0$, $\omega \left(x\right)\ne 0$ 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 open question posed by Zhou (2020), and in particular when $k=0$ it improves the result obtained by Bandle et al. (2000). An interesting fact is that in the case $\gamma >0$, the problem does not admit global solutions for any $p>1$ and ${\int }_{{\mathbb{R}}^N}\omega \left(x\right)dx>0.$

在本文中，我们研究了一个具有非局部非线性的非齐次伪抛物方程${你}_{吨}-ķ\Delta {你}_{吨}-\Delta 你={我}_{0+}^{\gamma }\left({\left|你\right|}^p\right)+\omega \left(X\right),(吨,X)\in (0,\infty )\times {\mathbb{R}}^{ñ},$在哪里$p>1,ķ\ge 0$,$\omega \left(X\right)\ne 0$和${我}_{0+}^{\gamma }$是阶的左黎曼-刘维尔分数积分$\gamma \in (0,1).$基于测试函数法，我们证明了危急情况的爆破结果$\gamma =0,p=p_C$为了$ñ\ge 3$，它回答了Zhou (2020) 提出的一个悬而未决的问题，尤其是当$ķ=0$它改进了 Bandle 等人获得的结果。（2000 年）。一个有趣的事实是，在这种情况下$\gamma >0$, 该问题不承认任何全局解决方案$p>1$和${\int }_{{\mathbb{R}}^{ñ}}\omega \left(X\right)dX>0.$