Reaction-diffusion fronts in inhomogeneous media
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In this thesis, we study the asymptotic behavior of solutions to the reaction-advection-diffusion equation ut = ∆zu + B(z, t) · ∇zu + f(u), z ∈ R n , t > 0 under various conditions on the prescribed flow B. Our goal is to characterize, bound, and compute the speed of propagating fronts that develop in the solution u and to describe their dependence on the flow B. We focus mainly on the case when f is the KPP nonlinearity f(u) = u(1 − u). In the first section, we consider the case that B is a temporally random field having a spatial shear structure and Gaussian statistics. We show that the solution to the initial value problem develops traveling fronts, almost surely, which are characterized by a deterministic variational principle. In the second section, we use this and other variational principles to derive analytical estimates on the speed of propagating fronts. In the final section, we use the variational principle to compute the front speed numerically. The mathematical analysis involves perturbation expansions, ergodic theorems, and techniques from the theory of large deviations. We use numerical methods for computing the principal Lyapunov exponents of parabolic operators, which appear in the variational characterization of the front speed.