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Register a new userApproximation of random diffusion equation by nonlocal diffusion equation in free boundary problems of one space dimension
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We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_epsilon(x)=frac 1epsilon J(frac xepsilon)$ for small $epsilon>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $epsilon$.
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We study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and reveal some fundamental differences from its one dimensional version considered in cite{cdjfa} recently. Technically, this high dimensional problem is much more difficult to treat since it involves two kernel functions which arise from the original kernel function $J(|x|)$ in rather implicit ways. By introducing new techniques, we are able to determine the long-time dynamics of the model, including firstly finding the threshold condition on the kernel function that governs the onset of accelerated spreading, and the determination of the spreading speed when it is finite. Moreover, for two important classes of kernel functions, sharp estimates of the spreading profile are obtained. More precisely, for kernel functions with compact support, we show that logarithmic shifting occurs from the finite wave speed propagation, which is strikingly different from the one dimension case; for kernel functions $J(|x|)$ behaving like $|x|^{-beta}$ for $xinR^N$ near infinity, we obtain the rate of accelerated spreading when $betain (N, N+1]$, which is the exact range of $beta$ where accelerated spreading is possible. These sharp estimates are obtained by constructing subtle upper and lower solutions, based on careful analysis of the involved kernel functions.
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We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and the geometry of nonlocal interaction neighborhoods. Such an approach can be potentially useful to address similar issues in peridynamics, smoothed particle hydrodynamics and other nonlocal models. We first show the well-posedness of the newly formulated nonlocal problems and then analyze their asymptotic convergence to the local limit as the nonlocality parameter shrinks to zero. We justify the second order localization rate, which is the optimal order attainable in the absence of physical boundaries.
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