In this paper, we consider a class of parabolic or pseudo-parabolic equation with nonlocal source term:

$$\begin{aligned} u_t-\nu \triangle u_t-\hbox {div}(\rho (|\nabla u|)^2\nabla u)=u^p(x,t)\int _{\Omega }k(x,y)u^{p+1}(y,t)dy, \end{aligned}$$

where \(\nu \ge 0\) and \(p>0\). Using some differential inequality techniques, we prove that blow-up cannot occur provided that \(q>p\), also, we obtain some finite-time blow-up results and the lifespan of the blow-up solution under some different suitable assumptions on the initial energy. In particular, we prove finite-time blow-up of the solution for the initial data at arbitrary energy level. Furthermore, the lower bound for the blow-up time is determined if blow-up does occur.

中文翻译：

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一类具有非局部源的抛物线方程或伪抛物线方程的爆破现象

在本文中，我们考虑一类具有非局部源项的抛物线或伪抛物线方程：

$$ \ begin {aligned} u_t- \ nu \ triangle u_t- \ hbox {div}（\ rho（| \ nabla u |）^ 2 \ nabla u）= u ^ p（x，t）\ int _ {\ Ω} k（x，y）u ^ {p + 1}（y，t）dy，\ end {aligned} $$

其中\（\ nu \ ge 0 \）和\（p> 0 \）。使用一些微分不等式技术，我们证明了在\（q> p \）的情况下不会发生爆炸，并且，在一些不同的适当假设下，我们获得了一些有限时间的爆炸结果和爆炸解决方案的寿命在初始能量上。特别是，我们证明了在任意能级下对初始数据的解的有限时间分解。此外，如果确实发生爆燃，则确定爆燃时间的下限。