2023/10/20 by M. Al Haj, Mohammad Al Haj, R. Monneau +1
Medicine · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #Mathematical Biology Tumor Growth
paper · doi:10.1016/j.jde.2023.10.017
We consider general fully nonlinear discrete reaction-diffusion equations u t = F [ u ] , described by some function F . In the positively monostable case, we study monotone traveling waves of velocity c , connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of F , we show that there is a minimal velocity c F + such that there is a branch of traveling waves with velocities c ≥ c F + and no traveling waves for c < c F + . We also show that the map F ↦ c F + is not continuous for the L ∞ norm on F . Assuming more regularity of F close to the unstable state 0, we show that c F + ≥ c F ⁎ , where the velocity c F ⁎ can be computed from the linearization of the equation around the unstable state 0. In addition, we show that the inequality can be strict for certain nonlinearities F . On the contrary, under a KPP condition on F , we show the equality c F + = c F ⁎ . Finally, we provide an example in which c F + is negative.