Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

TL;DR

This paper investigates the asymptotic behavior of blowing-up radial solutions for a class of quasilinear elliptic systems with gradient terms, providing precise boundary asymptotics and establishing a strong maximal principle.

Contribution

It introduces new asymptotic analysis for blowing-up solutions of quasilinear elliptic systems with gradient dependence, including a strong maximal principle and auxiliary system study.

Findings

Derived precise boundary asymptotics for solutions

Established a strong maximal principle for the system

Analyzed an auxiliary asymptotically autonomous system

Abstract

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: \begin{center} $\begin{cases} \Delta_p u= v^m| \nabla u |^\alpha& \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases}$ \end{center} where $\Omega\subset\mathbb{R}^N(N\geq 2)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1<p<\infty$, $m,q>0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $\delta:=(p-1-\alpha)(p-1-\beta)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that,we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\R^3$.