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Abstract:

In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. © 2010 Taylor & Francis.

Registro:

Documento: Artículo
Título:Nonlocal higher order evolution equations
Autor:Rossi, J.D.; Schönlieb, C.-B.
Filiación:Department de Matemáticas, FCEYN, Universidad de Buenos Aires, 1428 - Buenos Aires, Argentina
Institute for Numerical and Applied Mathematics, Georg-August University of Göttingen, Lotzestr. 16-18, D-37083 Göttingen, Germany
Palabras clave:Asymptotic behaviour; Higher order; Nonlocal diffusion
Año:2010
Volumen:89
Número:6
Página de inicio:949
Página de fin:960
DOI: http://dx.doi.org/10.1080/00036811003735824
Título revista:Applicable Analysis
Título revista abreviado:Appl. Anal.
ISSN:00036811
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00036811_v89_n6_p949_Rossi

Referencias:

  • Fife, P., (2003) Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions, Trends in Nonlinear Anal, , Springer, Berlin
  • Bates, P., Chen, X., Chmaj, A., Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions (2005) Calc. Var., 24, pp. 261-281
  • Bates, P., Fife, P., Ren, X., Wang, X., Travelling waves in a convolution model for phase transitions (1997) Arch. Rat. Mech. Anal, 138, pp. 105-136
  • Coville, J., On uniqueness and monotonicity of solutions on non-local reaction diffusion equations (2006) Ann. Mat. Pura Appl., 185, pp. 461-485
  • Coville, J., Maximum principles, sliding techniques and applications to nonlocal equations, Electronic (2007) J. Diff. Eqns., 2007 (68), pp. 1-23
  • Coville, J., Dávila, J., Martínez, S., Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity (2008) SIAM J. Math. Anal., 39 (5), pp. 1693-1709
  • Coville, J., Dupaigne, L., Propagation speed of travelling fronts in nonlocal reaction diffusion equations (2005) Nonlinear Anal., 60, pp. 797-819
  • Coville, J., Dupaigne, L., On a nonlocal equation arising in population dynamics (2007) Proc. Roy. Soc. Edinburgh, 137, pp. 1-29
  • Zhang, L., Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks (2004) J. Diff. Eqns., 197 (1), pp. 162-196
  • Bates, P., Chmaj, A., An integrodifferential model for phase transitions: Stationary solutions in higher dimensions (1999) J. Stat. Phys., 95, pp. 1119-1139
  • Bates, P., Chmaj, A., A discrete convolution model for phase transitions (1999) Arch. Rat. Mech. Anal., 150, pp. 281-305
  • Carrillo, C., Fife, P., Spatial effects in discrete generation population models (2005) J. Math. Biol, 50 (2), pp. 161-188
  • Chen, X., Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations (1997) Adv. Diff. Eqns., 2, pp. 125-160
  • Fife, P., Wang, X., A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions (1998) Adv. Diff. Eqns., 3 (1), pp. 85-110
  • Silvestre, L., Hölder estimates for solutions of integro differential equations like the fractional laplace (2006) Indiana Univ. Math. J., 55 (3), pp. 1155-1174
  • Wang, X., Metastability and stability of patterns in a convolution model for phase transitions (2002) J. Diff. Eqns., 183, pp. 434-461
  • Buades, A., Coll, B., Morel, J.-M., On image denoising methods (2005) SIAM Multiscale Model. Simulation, 4 (2), pp. 490-530
  • Kindermann, S., Osher, S., Jones, P., Deblurring and denoising of images by nonlocal functionals (2005) SIAM Multiscale Model. Simulation, 4 (4), pp. 1091-1115
  • Gilboa, G., Osher, S., (2006) Nonlocal Linear Image Regularization and Supervised Segmentation, , UCLA CAM Rep. No. 06-47, California
  • Fife, P., Models for phase separation and their mathematics, Electronic (2000) J. Diff. Eqns., 2000 (48), pp. 1-26
  • Bertozzi, A., Esedoglu, S., Gillette, A., Inpainting of binary images using the Cahn-Hilliard equation (2007) IEEE Trans. Image Proc., 16 (1), pp. 285-291
  • Bertozzi, A., Esedoglu, S., Gillette, A., Analysis of a two-scale Cahn-Hilliard model for image inpainting (2007) Multiscale Mod. Simul., 6 (3), pp. 913-936
  • Michelson, D., Steady solutions of the Kuramoto-Sivashinsky equation (1986) Physica D, 19, pp. 89-111
  • Cross, M.C., Hohenberg, P.C., Pattern formation outside of equilibrium (1993) Rev. Mod. Phys., 65 (3), pp. 851-1112
  • Polyanin, A.D., (2001) Handbook of Linear Partial Differential Equations for Engineers and Scientists, , Chapman and Hall/CRC, London
  • Happel, J., Brenner, H., (1973) Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Mechanics of Fluids and Transport Processes), , Springer, Berlin
  • Den Hartog, J.P., (1987) Advanced Strength of Materials, , Dover Publications, New York
  • Birkner, M., Lòpez-Mimbelab, J.A., Wakolbing, A., Comparison results and steady states for the Fujita equation with fractional Laplacian (2005) Ann. Inst. Henri Poincare (C) Non-Linear Anal., 22 (1), pp. 83-97
  • Fino, A., Karch, G., Decay of mass for nonlinear equation with fractional Laplacian Monatsh. Math., , (to appear) DOI: 10.1007/s00605-009-0093-3
  • Chen, W., Holm, S., Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency (2004) J. Acoust. Soc. Am., 115 (4), pp. 1424-1430
  • Han, J., The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation, Electronic (2004) J. Diff. Eqns., 2004 (113), pp. 1-9
  • Nishiura, Y., Ohnishi, I., Some mathematical aspects of the micro-phase separation in diblock copolymers (1995) Physica D, 84, pp. 31-39
  • Ohta, T., Kawasaki, K., Equilibrium morphology of block copolymer melts (1986) Macromolecules, 19, pp. 2621-2632
  • Chasseigne, E., Chaves, M., Rossi, J.D., Asymptotic behaviour for nonlocal diffusion equations (2006) J. Math. Pures Appl., 86, pp. 271-291
  • Ignat, L.I., Rossi, J.D., Refined asymptotic expansions for nonlocal evolution equations (2008) J. Evol. Eqns., 8 (4), pp. 617-629
  • Andreu, F., Mazon, J.M., Rossi, J.D., Toledo, J., The Neumann problem for nonlocal nonlinear diffusion equations (2008) J. Evol. Eqns., 8 (1), pp. 189-215
  • Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N., Boundary fluxes for non-local diffusion (2007) J. Diff. Eqns., 234, pp. 360-390
  • Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N., How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems (2008) Arch. Rat. Mech. Anal., 187 (1), pp. 137-156
  • Ignat, L.I., Rossi, J.D., A nonlocal convection-diffusion equation (2007) J. Funct. Anal., 251, pp. 399-437
  • Cortázar, C., Coville, J., Elgueta, M., Martínez, S., A non local inhomogeneous dispersal process (2007) J. Diff. Eqns., 241 (2), pp. 332-358
  • Cortazar, C., Elgueta, M., Rossi, J.D., A non-local diffusion equation whose solutions develop a free boundary (2005) Ann. Henri Poincaré, 6 (2), pp. 269-281

Citas:

---------- APA ----------
Rossi, J.D. & Schönlieb, C.-B. (2010) . Nonlocal higher order evolution equations. Applicable Analysis, 89(6), 949-960.
http://dx.doi.org/10.1080/00036811003735824
---------- CHICAGO ----------
Rossi, J.D., Schönlieb, C.-B. "Nonlocal higher order evolution equations" . Applicable Analysis 89, no. 6 (2010) : 949-960.
http://dx.doi.org/10.1080/00036811003735824
---------- MLA ----------
Rossi, J.D., Schönlieb, C.-B. "Nonlocal higher order evolution equations" . Applicable Analysis, vol. 89, no. 6, 2010, pp. 949-960.
http://dx.doi.org/10.1080/00036811003735824
---------- VANCOUVER ----------
Rossi, J.D., Schönlieb, C.-B. Nonlocal higher order evolution equations. Appl. Anal. 2010;89(6):949-960.
http://dx.doi.org/10.1080/00036811003735824