Learn More
We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative properties such as mono-tonicity and a priori integral inequalities. The computation uses an explicit(More)
We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present(More)
We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion(More)
We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in space-time random incompressible flows in dimension d > 1. We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time front speeds (spreading rates) are deterministic in all directions for compactly supported initial data.(More)
In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with(More)
We study the problem of estimating the coefficients in an ellip-tic partial differential equation using noisy measurements of a solution to the equation. Although the unknown coefficients may vary on many scales we aim only at estimating their slowly varying parts, thus reducing the complexity of the inverse problem. However, ignoring the fine-scale(More)
We consider solutions of an elliptic partial differential equation in R d with a stationary, random conductivity coefficient that is also periodic with period L. Boundary conditions on a square domain of width L are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary(More)
We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is either the ignition nonlinear-ity or the bistable nonlinearity, under suitable restrictions on the size of the spatial fluctuations. It is known that the solution develops an interface which(More)