We study the non-local reaction-diffusion system with Neumann boundary conditionswhere p, q > 0, u 0 (x), v 0 (x) ∈ C( ) are nonnegative and nontrivial functions, ∈ R N a bounded connected and smooth domain. We determine the existence and uniqueness of the solution. The blow-up phenomenon is considered and the blow-up rates are obtained.
We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole R N , N 1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In R N we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t → ∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t → ∞, they converge to zero.
We study the numerical approximation for the following non-local reaction-diffusion system u i (t) = N j=−N hJ(x i − x j)(u j (t) − u i (t)) + v p i (t) v i (t) = N j=−N hJ(x i − x j)(v j (t) − v i (t)) + u q i (t) u i (0) = u 0 (x i), v i (0) = v 0 (x i), in a bounded domain, where p, q > 0, −N ≤ i ≤ N and u 0 (x), v 0 (x) ∈ C(Ω) are nonnegative and nontrivial functions and Ω ⊂ R N is a bounded, connected and smooth domain. We prove the existence and uniqueness of solution if pq ≥ 1 or if one of the initials conditions is different from zero and pq < 1. We analyzed a comparison principle for the solutions. Finally, we study the convergence of the numerical scheme.
Abstract. Let J : R → R be a nonnegative, smooth function with R J(r)dr = 1, supported in [−1, 1], symmetric, J(r) = J(−r), and strictly increasing in [−1, 0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equationWe prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.
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