Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems withp-Laplacian with Nonlocal Sources

Abstract

This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

Blow-up for an evolution p-laplace system with nonlocal sources and inner absorptions

“…Motivated by the above works, in this paper, we investigate the blow-up properties of solutions of the problem (1.1) and extend the results of [4,11,16,19] to more generalized cases.…”

“…For other works on parabolic system like (1.1), we refer readers to [17][18][19][20][21][22][23][24][25][26][27][28][29][30] and the references therein.…”

Global existence and blow-up solution for doubly degenerate parabolic system with nonlocal sources and inner absorptions

“…For p‐Laplacian system, Cui and Yang and Li studied the following equations: …”

“…If f .u/ D u p , p > 1, the solution may blow up in finite time, see [12][13][14][15]. For p-Laplacian system, Cui and Yang [16] and Li [17] studied the following equations: 8 < :…”

Global existence and blow-up solutions for doubly degenerate parabolic system with nonlocal source

“…Furthermore, they obtained the blow-up set and asymptotic behavior provided that the solution blows up in finite time. For p-Laplacian systems, Cui and Yang [29] and Li [30] studied the following equations: Very recently, Zhang and Yang [31] further studied the solutions for system (1.1) with m = n = 1. They first got the non-existence result for a related elliptic systems of non-increasing positive solutions and by using this result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions were obtained.…”