In this work, we deal with the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term: $$ \textstyle\begin{cases} (b(u) )_{t} =\nabla \cdot ( \vert \nabla u \vert ^{p-2}\nabla u )+f(x,u, \vert \nabla u \vert ^{2},t) &\text{in } \varOmega \times (0,t^{*}), \\ \frac{\partial u}{\partial n}=0 &\text{on } \partial \varOmega \times (0,t^{*}),\\ u(x,0)=u_{0}(x)\geq 0 & \text{in } \overline{\varOmega }, \end{cases} $$ where $p>2$ , the spatial domain $\varOmega \subset \mathbb{R}^{N}$ ( $N\geq 2$ ) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time $t^{*}$ . The upper bounds of the blow-up time $t^{*}$ and the upper estimates of the blow-up rate are also obtained.

中文翻译：

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具有梯度源项的抛物型p- Laplacian方程的爆破分析。

在这项工作中，我们使用梯度源项处理以下抛物线p-Laplacian方程的爆炸解：$$ \ textstyle \ begin {cases（b（u））_ {t} = \ nabla \ cdot （\ vert \ nabla u \ vert ^ {p-2} \ nabla u）+ f（x，u，\ vert \ nabla u \ vert ^ {2}，t）＆\ text {in} \ varOmega \ times（ 0，t ^ {*}），\\ \ frac {\ partial u} {\ partial n} = 0＆\ text {on} \ partial \ varOmega \ times（0，t ^ {**）），\\ u （x，0）= u_ {0}（x）\ geq 0＆\ text {in} \ overline {\ varOmega}，\ end {cases} $$其中$ p> 2 $，空间域$ \ varOmega \子集\ mathbb {R} ^ {N} $（$ N \ geq 2 $）是有界的，并且边界Ω是平滑的。我们的研究依赖于创建一些合适的辅助函数以及使用微分不等式技术和抛物线最大原理。我们给出足够的条件以确保解决方案在有限的时间$ t ^ {*} $爆炸。