Can I read Global Behaviour of Radially Symmetric Solutions Stable at Infinity for Gradient Systems on EtoBox?

Global Behaviour of Radially Symmetric Solutions Stable at Infinity for Gradient Systems by Risler, Emmanuel is a scholarly article available to read on EtoBox.

What is Global Behaviour of Radially Symmetric Solutions Stable at Infinity for Gradient Systems about?

This paper is concerned with radially symmetric solutions of systems of the form \[ u_t = -\nabla V(u) + \Delta_x u \] where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions "stable at infinity", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+\infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).

Author

Risler, Emmanuel

Published

2017

Language

EN