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Asymptotic behavior for a nonlocal diffusion equation in exterior domains: the critical two-dimensional case. (English) Zbl 1332.35158

Summary: We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, \(\partial_t u = J \ast u - u\), where \(J\) is a smooth, radially symmetric kernel with support \(B_d(0) \subset \mathbb{R}^2\). The problem is set in an exterior two-dimensional domain which excludes a hole \(\mathcal{H}\), and with zero Dirichlet data on \(\mathcal{H}\). In the far field scale, \(\xi_1 \leq | x | t^{- 1 / 2} \leq \xi_2\) with \(\xi_1, \xi_2 > 0\), the scaled function \(\log t u(x, t)\) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by \(J\). The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic ’logarithmic momentum’ of the solution, \(\lim_{t \to \infty} \int_{\mathbb{R}^2} u(x, t) \log | x | d x\). This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, \(| x | \leq t^{1 / 2} h(t)\) with \(\lim_{t \to \infty} h(t) = 0\), the scaled function \(t(\log t)^2 u(x, t) / \log | x |\) converges to a multiple of \(\phi(x) / \log | x |\), where \(\phi\) is the unique stationary solution of the problem that behaves as \(\log | x |\) when \(| x | \to \infty\). The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, \(| x | \geq t^{1 / 2} g(t)\) with \(g(t) \to \infty\), the solution is proved to be of order \(o((t \log t)^{- 1})\).

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
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[1] Bates, P. W.; Chmaj, A., An integrodifferential model for phase transitions: stationary solutions in higher dimensions, J. Stat. Phys., 95, 5-6, 1119-1139, (1999) · Zbl 0958.82015
[2] Bates, P. W.; Chmaj, A., A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150, 4, 281-305, (1999) · Zbl 0956.74037
[3] Bates, P. W.; Zhao, G., Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332, 1, 428-440, (2007) · Zbl 1114.35017
[4] Brändle, C.; Chasseigne, E.; Quirós, F., Phase transitions with midrange interactions: a nonlocal Stefan model, SIAM J. Math. Anal., 44, 4, 3071-3100, (2012) · Zbl 1387.35594
[5] Carrillo, C.; Fife, P., Spatial effects in discrete generation population models, J. Math. Biol., 50, 2, 161-188, (2005) · Zbl 1080.92054
[6] Chasseigne, E.; Chaves, M.; Rossi, J. D., Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86, 3, 271-291, (2006) · Zbl 1126.35081
[7] Cortázar, C.; Elgueta, M.; Quirós, F.; Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation in domains with holes, Arch. Ration. Mech. Anal., 205, 2, 673-697, (2012) · Zbl 1288.76074
[8] Cortázar, C.; Elgueta, M.; Quirós, F.; Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation on the half line, Discrete Contin. Dyn. Syst., 35, 4, 1391-1407, (2015) · Zbl 1310.35236
[9] Cortázar, C.; Elgueta, M.; Quirós, F.; Wolanski, N., Asymptotic behavior for a one-dimensional nonlocal diffusion equation in exterior domains, preprint · Zbl 1332.35158
[10] Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, (Trends in Nonlinear Analysis, (2003), Springer-Verlag Berlin), 153-191 · Zbl 1072.35005
[11] Gilboa, G.; Osher, S., Nonlocal operators with application to image processing, Multiscale Model. Simul., 7, 3, 1005-1028, (2008) · Zbl 1181.35006
[12] Herraiz, L., Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 1, 49-105, (1999) · Zbl 0918.35025
[13] Ignat, L. I.; Rossi, J. D., Refined asymptotic expansions for nonlocal diffusion equations, J. Evol. Equ., 8, 4, 617-629, (2008) · Zbl 1178.35128
[14] Terra, J.; Wolanski, N., Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data, Discrete Contin. Dyn. Syst., 31, 2, 581-605, (2011) · Zbl 1222.35107
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