This paper deals with the blow‐up phenomenon to the following quasi‐linear pseudo‐parabolic equation with nonlocal source:

$$u_t-\mathrm{\Delta }{u}_t-\nabla ·(|\nabla u{|}^{2q}\nabla u)=u^p(x,t){\int }_{\mathrm{\Omega }}K(x,y)u^{p+1}(y,t)dy,x,y\in \mathrm{\Omega },t>0,$$

where $\mathrm{\Omega }\subseteq {ℝ}^n,n\ge 3$, is a bounded domain with smooth boundary. Here, 0 < q ≤ p and K(x,y) is an integrable real‐valued function. We show that for q < p, the blow‐up occurs in finite time with suitable initial data and arbitrary positive initial energy. We also state some key results based on the conception of limiting the energy function in the case of nonnegative initial energy. Besides, we obtain the exact blow‐up time under some certain conditions.

中文翻译：

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一类带有非局部源的伪抛物方程的爆破现象和确切的爆破时间

本文针对以下带有非局部源的拟线性拟抛物方程的爆炸现象：

$${ü}_{Ť}-\mathrm{\Delta }{ü}_{Ť}-\nabla ·（|\nabla ü{|}^{2q}\nabla ü）={ü}^p（X，Ť）{\int }_{\mathrm{\Omega }}ķ（X，ÿ）{ü}^{p+\mathrm{1个}}（ÿ，Ť）dÿ，X，ÿ\in \mathrm{\Omega }，Ť>0，$$

哪里 $\mathrm{\Omega }\subseteq {ℝ}^{ñ}，ñ\ge 3$是具有平滑边界的有界域。在此，0 <  q  ≤  p和ķ（X，ÿ）是积实值函数。我们证明对于q  <  p，爆炸发生在有限的时间内，具有合适的初始数据和任意正的初始能量。我们还基于在非负初始能量的情况下限制能量函数的概念陈述了一些关键结果。此外，在某些条件下，我们可以获得确切的爆炸时间。