2023/06/07 by Ramon Oliver-Bonafoux, Oliver-Bonafoux, Ramon, Emmanuel Risler +1
Engineering · Mathematics · Medicine · #35B38 #35B40 #35K57 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2306.04413
openalex publication_date 2023/06/07 · openalex created_date 2023/06/09 · openalex updated_date 2026/07/28
This article addresses the issue of global convergence towards pushed travelling fronts for solutions of parabolic systems of the form ut = - ∇ V(u) + uxx , where the potential V is coercive at infinity. It is proved that, if an initial condition x↦ u(x,t=0) approaches, rapidly enough, a critical point e of V to the right end of space, and if, for some speed c0 greater than the linear spreading speed associated with e, the energy of this initial condition in a frame travelling at the speed c0 is negative \unicodex2013 with symbols, ∫ℝ ec0 x((1)/(2) ux(x,0)2 + V(u(x,0))- V(e)) dx lt; 0 , then the corresponding solution invades e at a speed c greater than c0, and approaches, around the leading edge and as time goes to +∞, profiles of pushed fronts (in most cases a single one) travelling at the speed c. A necessary and sufficient condition for the existence of pushed fronts invading a critical point at a speed greater than its linear spreading speed follows as a corollary. In the absence of maximum principle, the arguments are purely variational. The key ingredient is a Poincaré inequality showing that, in frames travelling at speeds exceeding the linear spreading speed, the variational landscape does not differ much from the case where the invaded equilibrium e is stable. The proof is notably inspired by ideas and techniques introduced by Th. Gallay and R. Joly, and subsequently used by C. Luo, in the setting of nonlinear damped wave equations.