A Compactness Tool for the Analysis of Nonlocal Evolution Equations

Abstract

In this paper we analyze the long time behavior of the solutions of a nonlocal diffusionconvection equation. We give a new compactness criterion in the Lebesgue spaces L p ((0, T ) × Ω) and use it to obtain the first term in the asymptotic expansion of the solutions. Previous results of [J. Bourgain, H. Brezis, and P. Mironescu, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001] are used to obtain a compactness result in the spirit of the AubinLions-Simon lemma. COMPACTNESS TOOLS FOR NONLOCAL EVOLUTION EQUATIONSSince J and G have mass one the mass conservation property holds:Since the proof of the global well-posedness follows by the same fixed-point argument as in [15] we will omit it here.Observe that under the assumptions (H0), (H1), and (H3) there exist positive constants R, δ such that|ξ| ≥ R, Downloaded 04/13/15 to 128.122.253.212. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php When q = 1 + 1/d and B = 0 1,d , U is the unique solution of the following equation:The well-posedness of this system has been analyzed in [8] in the one-dimensional case and in [9] in the multidimensional case. It has been proved in [1] that the profile f m is of constant sign and decays exponentially to zero as |x| → ∞.We remark that in the case of the symmetric function G, i.e., G(z) = G(−z), the solution of (1.1) converges to the heat kernel since in this case B vanishes. When B = 0 1,d we obtain in the limit the solutions of the viscous convection-diffusion equation. Throughout the paper we will consider the case of nonnegative initial data, so nonnegative solutions of system (1.1). The case of sign-change solutions could also be analyzed with small modifications of the proof (see [8] for a rigorous treatment of the critical case for the convection-diffusion equation).In the linear case, i.e., u t = J * u − u, the asymptotic behavior has been obtained in [5] by means of Fourier analysis techniques and in [19] by scaling methods. In [19] the scaling argument works since it is applied to the smooth part of the solution. Refined asymptotics have been obtained in [16,18]. We also recall here [31,32], where a scaling method is used for equations of the type u t = J * u − u − u p . There the authors obtain barriers for W and its derivatives, W being the smooth part of the solution of the linear equation u t = J * u − u. Once these barriers are obtained the authors split the solution of the nonlinear problem in a way that permits obtaining uniform Hölder estimates and then compactness. The method developed here is more flexible in the sense that it uses only energy estimates that involve the linear part of the equation and the good sign of the nonlinearity.In the local case, i.e., u t = Δu + a · ∇(|u| q−1 u), the same analysis has been performed in a series of papers. In [10] the case q ≥ 1 + 1/d is treated and the results in the critical case have been obtained by a careful space-time change of variables and using weighted Sobolev spaces. The subcritical...

Asymptotic behavior of solutions to fractional diffusion–convection equations

“…There are situations when the convection is also nonlocal, . We refer to for the supercritical case and for the critical case . However, for the subcritical case, that is, there are no results on the long time behavior of the solutions. (iii)The case of nonlinear local diffusion also brings considerable difficulties, for instance for porous‐medium type diffusion and convection the model becomes .…”

“…. We refer to [27] for the supercritical case q > 1 + 1/N and [25] for the critical case q = 1 + 1/N . However, for the subcritical case, that is, q < 1 + 1/N there are no results on the long time behavior of the solutions.…”

Nonlocal evolution equations in perforated domains

“…For random homogenization of an obstacle problem we cite the work of Caffarelli and Mellet. 18 For nonlocal evolution problems with smooth kernels we refer to other works [19][20][21][22][23][24][25] and the references therein. Some applications of this kind of equations may be seen, for instance, in other works [26][27][28][29][30][31] where models to peridynamics, elasticity, population dynamic, biology, etc are considered.…”

Nonlocal problems in perforated domains

“…On the other hand, nonlocal equations with non-singular kernel attracted some attention recently, see [2,16,21,22,23,35] for a non-exhaustive list of references. We also mention [31,32] where asymptotic problems in such nonlocal equations have been recently studied.…”