2015/03/25 by Andrej Zlatoš, Zlatos, Andrej
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.1503.07599
openalex publication_date 2015/03/25 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We study reaction-diffusion equations in one spatial dimension and with\ngeneral (space- or time-) inhomogeneous mixed bistable-ignition reactions. For\nthose satisfying a simple quantitative hypothesis, we prove existence and\nuniqueness of transition fronts, as well as convergence of "typical" solutions\nto the unique transition front (the existence part even extends to mixed\nbistable-ignition-monostable reactions). These results also hold for all pure\nignition reactions without any other hypotheses, but not for all pure bistable\nreactions. In fact, we find examples of either spatially or temporally periodic\npure bistable reactions (independent of the other space-time variable) for\nwhich we can prove non-existence of transition fronts. These are the first such\nresults for periodic media which are non-degenerate in a natural sense, and the\nspatially periodic examples also prove a conjecture from citeDHZ.\n