## 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
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.