Browsing by Subject "Reaction-diffusion equations"

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    Reaction-diffusion fronts in inhomogeneous media
    (2006) Nolen, James Hilton; Xin, Jack; Souganidis, Panagiotis
    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.
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    Singular limits of reaction diffusion equations of KPP type in an infinite cylinder
    (2007) Carreón, Fernando; Souganidis, Takis
    In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case.