Global behaviour of radially symmetric solutions stable at infinity for gradient systems

Abstract

This paper is concerned with radially symmetric solutions of systems of the form \[ u_t = -\nabla V(u) + Δ_x u \] 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).

71 pages, 15 figures. arXiv admin note: text overlap with arXiv:1604.02002, arXiv:1703.01221, arXiv:1604.00804

Cited by in corpus (2)

• Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension
• Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems