We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert ^{q-2}s,$ $q>p$ and with bad sign at infinity as $\left\vert s\right\vert \rightarrow \infty $. We show the existence of locally-defined strong solutions to the problem with any initial condition $u_{0}\in L^{r}(\Omega )$ where $r\geq 2$ satisfies $r>N(q-p)/sp$. Then, we prove that finite time blow-up is possible for these problems in the range of parameters provided for $r,p,q$ and the initial datum $u_0$.

中文翻译：

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关于一些与分数 $p$-Laplacian 相关的退化非局部抛物线方程

我们考虑与分数 $% p $-Laplace 算子相关的退化抛物线方程 $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \ left( 0,1\right) $) 和像 $\left\vert s\right\vert ^{q-2}s,$ $q>p$ 一样增长的单调扰动，并且在无穷远处的坏符号为 $\left \vert s\right\vert \rightarrow \infty $. 我们展示了对具有任何初始条件 $u_{0}\in L^{r}(\Omega )$ 的问题的局部定义强解的存在，其中 $r\geq 2$ 满足 $r>N(qp)/ sp$。然后，我们证明了在为 $r,p,q$ 和初始数据 $u_0$ 提供的参数范围内这些问题的有限时间爆炸是可能的。