Some new results on reaction-diffusion equations and geometric flows
Cecilia De Zan · Padua Research Archive (University of Padova) · 2012
In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion equations in the unbounded domain Rn × (0 + ∞), in the cases when such a behavior is described in terms of moving interfaces. As first class of asymptotic problems we consider the singular limit of bistable reaction-diffusion equations in the case when the velocity of the traveling wave equation depends on the space variable, i.e. cε = cε(x), and it satisfies, in some suitable sense, cε/ετ → α, as ε → 0+, where α is a discontinuous function and τ is an integer that can be equal to 0 or 1. The second part of the thesis concerns semilinear reaction-diffusion equations with diffusion term of type tr(Aε(x)D2uε), where tr denotes the trace operator, Aε = σεσtε for some matrix map σε : Rn → Rn×(m+n) and Aε converges to a degenerate matrix. In order to establish such results rigorously, we modify and adapt to our problems the ”geometric approach” introduced by G. Barles and P. E. Souganidis for solving problems in Rn, and then partially revisited by G. Barles and F. Da Lio for reaction-diffusion equations in bounded domains. When it is possible we always consider the question of the well posedness of the Cauchy problems governing the motion of the fronts that describe the asymptotics we consider