2017/03/06 by Emmanuel Risler, Risler, Emmanuel
Engineering · Mathematics · Medicine · #35B38 #35B40 #35K57 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1703.02134
openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with radially symmetric solutions of systems of the form ut = -∇ V(u) + Δx 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 +∞, 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).