Descripción: We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions. © 2007.

Descripción: In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J * u - u + G * (f (u)) - f (u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut = Δ u + b ṡ ∇ (f (u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t → ∞ when f (u) = | u |q - 1 u with q > 1. We find the decay rate and the first-order term in the asymptotic regime. © 2007 Elsevier Inc. All rights reserved.

Descripción: In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. © 2008 Elsevier Masson SAS. All rights reserved.

Descripción: We study the asymptotic behavior for nonlocal diffusion models of the form ut = J * u - u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If over(J, ̂) (ξ) = 1 - A | ξ |α + o (| ξ |α) (0 < α ≤ 2), the asymptotic behavior is the same as the one for solutions of the evolution given by the α / 2 fractional power of the Laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is yet a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition. © 2006 Elsevier SAS. All rights reserved.

Descripción: In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut (x, t) = ∫Rd G (x - y) (u (y, t) - u (x, t)) d y. For example, we will consider equations like,ut (x, t) = under(∫, Rd) J (x, y) (u (y, t) - u (x, t)) d y + f (u) (x, t), and a nonlocal analogous to the p-Laplacian,ut (x, t) = under(∫, Rd) J (x, y) | u (y, t) - u (x, t) |p - 2 (u (y, t) - u (x, t)) d y . The energy method developed here allows us to obtain decay rates of the form,{norm of matrix} u (ṡ, t) {norm of matrix}Lq (Rd) ≤ C t- α, for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration. © 2009 Elsevier Masson SAS. All rights reserved.

Descripción: We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition. © 2006 Elsevier Inc. All rights reserved.

Descripción: In this work we consider the nonlocal stationary nonlinear problem (J * u)(x) - u(x) = -λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u(x) = 0 in ℝN Ω and p > 1. The kernel J involved in the convolution (J * u)(x) = ∫ℝN J(x - y)u(y) dy is a smooth, compactly supported nonnegative function with unit integral, while the weight a(x) is assumed to be nonnegative and is allowed to vanish in a smooth subdomain Ω0 of Ω. Both when a(x) is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter λ.

Descripción: 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 equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We 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. © 2007 American Mathematical Society.

Descripción: We study the large time behavior of nonnegative solutions of the Cauchy problem ut =R J(x,y)(u(y; t),u(x; t)) dy,up, u(x; 0) = u0(x) 2 L∞, where |x|αu0(x) → A < 0 as |x| → 1. One of our main goals is the study of the critical case p = 1+2=ff for 0 > ff > N, left open in previous articles, for which we prove that tff=2ju(x; t) , U(x; t)j → 0 where U is the solution of the heat equation with absorption with initial datum U(x; 0) = CA;N |x|,α. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data u0 in the supercritical case and also in the critical case (p = 1 + 2=N) for bounded and integrable u0.

Descripción: We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(s, ε)), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and ∫ β (s) d s = M. The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε → 0) and we show that limit functions are solutions to the two phase free boundary problem:Δ u = f χ{u ≢ 0} in  Ω {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on  Ω ∩ ∂ {u > 0}, where f = lim fε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case fε ≡ 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary. © 2006 Elsevier SAS. All rights reserved.