Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

Abstract

We consider a nonlocal parabolic equation associated with the fractional p-laplace operator, which was studied by Gal and Warm in [On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ., 14(1): 47-77, 2017]. By exploiting the boundary condition and the variational structure of the equation, according to the size of the initial dada, we prove the finite time blow-up, global existence, vacuum isolating phenomenon of the solutions. Furthermore, the upper and lower bounds of the blow-up time for blow-up solutions are also studied. The results generalize the results got by Gal and Warm.

Global well‐posedness and finite time blow‐up for a class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearity

“…We point out that, the literature on parabolic equations involving fractional 𝑝-Laplacian in bounded or unbounded domains with polynomial type nonlinearities is very large and rich. Here, we just refer the readers to [6,7,22,24,30,38] and the references cited there. For instance, Fu and Pucci [45], by using the potential well theory, proved the existence of global weak solutions and established the vacuum isolating and blow-up of strong solutions for the following problem:…”

“…We point out that, the literature on parabolic equations involving fractional p ‐Laplacian in bounded or unbounded domains with polynomial type nonlinearities is very large and rich. Here, we just refer the readers to [6, 7, 22, 24, 30, 38] and the references cited there. For instance, Fu and Pucci [45], by using the potential well theory, proved the existence of global weak solutions and established the vacuum isolating and blow‐up of strong solutions for the following problem: $$$$\begin{equation*} \hspace*{147pt}{\left\lbrace \def\eqcellsep{&}\begin{array}{l}u_{t}+ (-\Delta )^{s} u=|u|^{p-2}u, \;\; x\in \Omega , \; t&gt;0, \\[3pt] u(x,t)=0, \;\;\;\; x\in \mathbb {R}^{N}\backslash \Omega , \;t&gt; 0, \\[3pt] u(x,0)=u_{0}(x), \;\;\; x\in \Omega , \end{array} \right.$$…”

Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity

“…We point out that in the last years many authors have obtained important results on the fractional p−Laplacian in bounded or unbounded domains with polynomial type nonlinearities, for example see ( [17], [22], [4], [5], [33]) and the references therein. With the help of potential well theory, Fu and Pucci [42], studied the existence of global weak solutions and established the vacuum isolating and blow-up of strong solutions for the following class of problem…”

Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian

“…We point out that in the last years many authors have obtained important results on the fractional p−Laplacian in bounded or unbounded domains, for example see ( [21], [29], [6], [7], [21]) and the references therein. Mazón, Rossi and Toledo [20] considered a model of fractional diffusion involving a nonlocal version of the p-Laplacian operator…”