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

Abstract

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: $$ \begin{cases} \Delta_p u= v^m| \nabla u |^\alpha & \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\ (N\geq 1)$ 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 $\mathbb{R}^3$.