Stability of positive radial steady states for the parabolic H\'enon-Lane-Emden system

TL;DR

This paper extends the stability results of positive radial steady states from the nonlinear heat equation to the more complex parabolic Hénon-Lane-Emden system, showing all such states are stable under certain conditions.

Contribution

It generalizes stability analysis from a single equation to a coupled reaction-diffusion system with weighted nonlinearities, covering a broader class of solutions.

Findings

All positive radial steady states share the same asymptotic behavior at infinity.

These steady states are proven to be stable solutions in the entire space.

The results apply when the parameters lie on or above the Joseph-Lundgren critical curve.

Abstract

When it comes to the nonlinear heat equation $u_t - \Delta u = u^p$, the stability of positive radial steady states in the supercritical case was established in the classical paper by Gui, Ni and Wang. We extend this result to systems of reaction-diffusion equations by studying the positive radial steady states of the parabolic H\'enon-Lane-Emden system $$\left\{ \begin{aligned} u_t - \Delta u &= |x|^k v^p &\mbox{ in } \mathbb R^n \times (0,\infty),\\ v_t - \Delta v &= |x|^l u^q &\mbox{ in } \mathbb R^n \times (0,\infty), \end{aligned} \right.$$ where $k,l\geq 0$, $p,q\geq 1$ and $pq>1$. Assume that $(p,q)$ lies either on or above the Joseph-Lundgren critical curve which arose in the work of Chen, Dupaigne and Ghergu. Then all positive radial steady states have the same asymptotic behavior at infinity, and they are all stable solutions of the parabolic H\'enon-Lane-Emden system in $\mathbb R^n$.