Asymptotic behavior for nonlocal diffusion equations. (English) Zbl 1126.35081
The authors study the asymptotic behavior for nonlocal diffusion problems of the form \(u_t=J*u-u\), in the whole \(\mathbb R^n\) or in a bounded smooth domain with Dirichlet or Neumann conditions. In the case of \(\mathbb R^n\) the main result is that the long time behavior of 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. For the Dirichlet problem the authors prove that the asymptotic behavior of solutions shows exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile of an eigenfunction of the first eigenvavalue. For the Neumann problem the authors find an exponential convergence to the mean value of the initial condition.
Reviewer: Panagiotis Ch. Tsamatos (Ioannina)
MSC:
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35A15 | Variational methods applied to PDEs |
| 35R10 | Partial functional-differential equations |
Keywords:
nonlocal diffusion; fractional Laplacian; Dirichlet boundary conditions; Neumann boundary conditionsReferences:
| [1] | Bates, P.; Chmaj, A., An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95, 1119-1139 (1999) · Zbl 0958.82015 |
| [2] | Bates, P.; Chmaj, A., A discrete convolution model for phase transitions, Arch. Rat. Mech. Anal., 150, 281-305 (1999) · Zbl 0956.74037 |
| [3] | Bates, P.; Fife, P.; Ren, X.; Wang, X., Travelling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal., 138, 105-136 (1997) · Zbl 0889.45012 |
| [4] | Brezis, H., Analyse fonctionelle (1983), Masson: Masson Paris · Zbl 0511.46001 |
| [5] | Brezis, H.; Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62, 73-97 (1983) · Zbl 0527.35043 |
| [6] | Carrillo, C.; Fife, P., Spatial effects in discrete generation population models, J. Math. Biol., 50, 2, 161-188 (2005) · Zbl 1080.92054 |
| [7] | Cazenave, T.; Haraux, A., An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13 (1998), Clarendon Press: Clarendon Press Oxford, p. xiv · Zbl 0926.35049 |
| [8] | Chasseigne, E.; Vazquez, J. L., Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164, 2, 133-187 (2002) · Zbl 1018.35048 |
| [9] | Chen, X., Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations, 2, 125-160 (1997) · Zbl 1023.35513 |
| [10] | Cordoba, A.; Cordoba, D., A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249, 511-528 (2004) · Zbl 1309.76026 |
| [11] | C. Cortazar, M. Elgueta, J.D. Rossi, N. Wolanski, Boundary fluxes for non-local diffusion, Preprint; C. Cortazar, M. Elgueta, J.D. Rossi, N. Wolanski, Boundary fluxes for non-local diffusion, Preprint · Zbl 1113.35101 |
| [12] | Duoandikoetxea, J.; Zuazua, E., Moments, masses de Dirac et decomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315, 6, 693-698 (1992) · Zbl 0755.45019 |
| [13] | Durrett, R., Probability: Theory and Examples (1995), Duxbury Press |
| [14] | Escobedo, M.; Zuazua, E., Large time behavior for convection-diffusion equations in RN, J. Funct. Anal., 100, 1, 119-161 (1991) · Zbl 0762.35011 |
| [15] | Feller, W., An Introduction to the Probability Theory and Its Application, vol. 2 (1996), Wiley: Wiley New York |
| [16] | Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, (Trends in Nonlinear Analysis (2003), Springer-Verlag: Springer-Verlag Berlin), 153-191 · Zbl 1072.35005 |
| [17] | Fife, P.; Wang, X., A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Adv. Differential Equations, 3, 1, 85-110 (1998) · Zbl 0954.35087 |
| [18] | Körner, T. W., Fourier Analysis (1988), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0649.42001 |
| [19] | E. Lesigne, M. Peigné, Personal communication; E. Lesigne, M. Peigné, Personal communication |
| [20] | L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., submitted for publication; L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., submitted for publication · Zbl 1101.45004 |
| [21] | Wang, X., Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183, 434-461 (2002) · Zbl 1011.35073 |
| [22] | Zhang, L., Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations, 197, 1, 162-196 (2004) · Zbl 1054.45005 |
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