Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

Abstract

In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.

Nonexistence of global solutions for a class of viscoelastic wave equations

“…Two notable first articles on this subject are [30,29], see also [7,8,12,13,15,19,20,27,28], [31]- [34]. Related and interesting results have been reported in [2]- [5], [14], [16]- [18], [21,24,26,35]. The purpose of this work is to give sufficient conditions to get blow-up for the viscoelastic problem (P) with any positive value of the initial energy and improve the previous results.…”

Remarks on Blow-Up Phenomena in $p$-Laplacian Heat Equation with Inhomogeneous Nonlinearity

“…The question of blow-up for nonnegative classical solutions of p-Laplacian heat equations with various boundary conditions has attracted considerable attention in the mathematical community in recent years. See for instance [7][8][9][10].…”

A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

“…Our main technical tool is the theory of potential wells (see e.g. [5,7,8,12,13,17,21,24,26,27,29,31]) with a slight modification, which plays an essential role in the proofs of main results.…”