The subject of this thesis is the asymptotic behavior of a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion. We show that solutions of this equation asymptotically converge to the stationary states of the nonlinearity in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality. This result is an extension of earlier work by Evans and Souganidis and Majda and Souganidis where the Laplacian takes the place of the nonlocal diffusion. In Chapter 3, we will obtain this result for the nonlocal KPP equation in a periodic or almost-periodic medium by making use of the theory of periodic homogenization and adapting it to the nonlocal setting. In Chapter 4, we will prove that this result holds in a general stationary ergodic medium by obtaining a priori estimates for solutions of the cell problem and adapting the theory of stochastic homogenization of Hamilton-Jacobi equations.
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