Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy

Guangyu Xu,Jun Zhou

doi:10.3934/cpaa.2018086

Abstract

In this paper, we revisit the singular Non-Newton polytropic filtration equation, which was studied extensively in the recent years. However, all the studies are mostly concerned with subcritical initial energy, i.e., \begin{document} $E(u_0) , where \begin{document} $E(u_0)$ \end{document} is the initial energy and \begin{document} $d$ \end{document} is the mountain-pass level. The main purpose of this paper is to study the behaviors of the solution with \begin{document} $E(u_0)≥d$ \end{document} by potential well method and some differential inequality techniques.

Highlights

In this paper, we study the following Non-Newton polytropic filtration equation |x|−sut − ∇ · |∇um|p−2∇um = uq,(x, t) ∈ Ω × (0, T ), u(x, t) = 0,(x, t) ∈ ∂Ω × (0, T ), (1)u(x, 0) = u0(x), x ∈ Ω, where u0(x) is a nonnegative and nontrivial function, T ∈ (0, +∞] is the maximal existence time of solution, Ω is a bounded domain in RN (N > p) with smooth boundary ∂Ω, and the parameters in problem (1) satisfy m ≥ 1, p ≥ 2, 0 ≤ s ≤ 1 +, m m(N p − N + p) (2) mp − m < q ≤ N −p
U(x, 0) = u0(x), x ∈ Ω, where u0(x) is a nonnegative and nontrivial function, T ∈ (0, +∞] is the maximal existence time of solution, Ω is a bounded domain in RN (N > p) with smooth boundary ∂Ω, and the parameters in problem (1) satisfy m ≥ 1, p ≥ 2, 0 ≤ s ≤ 1 +, m m(N p − N + p) mp − m < q ≤
We define the sets related to global existence and blow-up as follows: Σ1 := {u ∈ Q : 0 < E(u) < d, H(u) > 0} ∪ {0}, Σ2 := {u ∈ Q : 0 < E(u) < d, H(u) < 0}, S :=

Summary

Introduction

We define the sets related to global existence and blow-up as follows: Σ1 := {u ∈ Q : 0 < E(u) < d, H(u) > 0} ∪ {0}, Σ2 := {u ∈ Q : 0 < E(u) < d, H(u) < 0} , S := Let u(t) be the weak solution of problem (1), we have the following conclusions: (i) If u0 ∈ Σ1, u(t) exists globally. For any t0 ∈ [0, T ), if E(u(t0)) ≤ 0, the weak solution of problem (1) blows up at some finite time T .

Objectives

Results

Conclusion