Existence of a pulled or pushed travelling front invading a critical point for parabolic gradient systems

2023/12/16 by Oliver-Bonafoux, Ramon, Risler, Emmanuel
#35B38 #35B40 #35K57 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.10427

Abstract

For nonlinear parabolic gradient systems of the form u_t = -∇ V(u) + u_xx , where the spatial domain is the whole real line, the state variable u is multidimensional, and the potential function V is coercive at infinity, the following result is proved: for every critical point of V which is not a global minimum point, there exists a travelling front, either pushed or pulled, invading this critical point at a speed which is not smaller than its linear spreading speed. By contrast with previous existence results of the same kind, no further assumption is made (neither that the invaded critical point is a non-degenerate local minimum point, nor other assumptions ensuring pushed invasion).

Related

• Global convergence towards pushed travelling fronts for parabolic gradient systems
• Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case
• Universal selection of pulled fronts
• Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum
• Global behaviour of bistable solutions for gradient systems in one unbounded spatial dimension