Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids. (Q2148415)

Let \(B_R\subset\mathbb{R}^N\) be the \(N\)-dimensional open ball centered in \(0\) with radius \(R>0\), let \(p>1\), \(m,q>0\), \(\alpha\geq 0\), \(\beta\in[0,m]\), and put \[\delta:=(p-1-\alpha)(p-1-\beta)-mq,\ \ \ \ \delta^*:=\beta+1-p(m+1).\] Consider the quasilinear elliptic system \[\left\{\begin{array}{ll}\Delta_p u=v^m|\nabla u|^\alpha,\ \ \ &{\text in} \ B_R,\\[3mm]\Delta_p v=v^\beta|\nabla u|^q,&{\text in}\ B_R.\end{array}\right.\hspace{3cm}(1)\] It is known that system \((1)\) admits unbounded positive radial solutions \((u,v)\) if \(\delta<0\), and one has:\begin{itemize}\item[-]\ \ \(\displaystyle{\lim_{t\rightarrow R^-}u(t)<\infty,\ \ \lim_{t\rightarrow R^-}v(t)=\infty}\) \ if and only if \ \(\delta<\delta^*\);\item[-]\ \ \(\displaystyle{\lim_{t\rightarrow R^-}u(t)=\infty,\ \ \lim_{t\rightarrow R^-}v(t)=\infty}\) \ if and only if \ \(\delta^*\leq\delta<0\).\end{itemize} In this paper, the exact asymptotic behavior of such solutions as \(t\rightarrow R^-\) when \(R=1\) is established (the result obtained being extendable to any \(R>0\) thanks to the homogeneity property of \((1)\)). It particular, the authors prove that if \((u,v)\) is an unbounded positive radial solution to \((1)\) with \(R=1\), then \[u'(t)\sim\frac{\lambda}{(1-t)^{\sigma_*}},\ \ \ \ \ v(t)\sim\frac{\mu}{(1-t)^{k_*}},\ \ \ {\text as}\ \ \ t\rightarrow 1^-,\] where \[\sigma_*=\frac{\delta^*}{\delta},\ \ \ \ \ k_*=\frac{p(p-1-\alpha)+q}{\delta}\] and \(\lambda,\mu\) are positive coefficients (explicitly computed) depending on \(p,q,\alpha,\beta,m\). To obtain this result, a strong maximal principle is proved and techniques from the theory of 3-dimensional cooperative systems are used.