Artículo

Este artículo es de Acceso Abierto y puede ser descargado en su versión final desde nuestro repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

In this work we consider the nonlocal stationary nonlinear problem (J * u)(x) - u(x) = -λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u(x) = 0 in ℝN \\ Ω and p > 1. The kernel J involved in the convolution (J * u)(x) = ∫ℝN J(x - y)u(y) dy is a smooth, compactly supported nonnegative function with unit integral, while the weight a(x) is assumed to be nonnegative and is allowed to vanish in a smooth subdomain Ω0 of Ω. Both when a(x) is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter λ.

Registro:

Documento: Artículo
Título:A logistic equation with refuge and nonlocal diffusion
Autor:García-Melián, J.; Rossi, J.D.
Filiación:Dpto. de Análisis Matemático, Universidad de la Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna, Spain
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 - Buenos Aires, Argentina
Palabras clave:Logistic problems; Nonlocal diffusion
Año:2009
Volumen:8
Número:6
Página de inicio:2037
Página de fin:2053
DOI: http://dx.doi.org/10.3934/cpaa.2009.8.2037
Título revista:Communications on Pure and Applied Analysis
Título revista abreviado:Commun. Pure Appl. Anal.
ISSN:15340392
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_15340392_v8_n6_p2037_GarciaMelian.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15340392_v8_n6_p2037_GarciaMelian

Referencias:

  • Andreu, F., Mazon, J.M., Rossi, J.D., Toledo, J., The Neumann problem for nonlocal nonlinear diffusion equations (2008) J. Evol. Eqns, 8, pp. 189-215
  • Bachman, G., Narici, L., (2000) Functional Analysis, , Dover, New York
  • 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., Chmaj, A., An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions (1999) J. Stat. Phys, 95, pp. 1119-1139
  • Bates, P., Fife, P., Ren, X., Wang, X., Traveling waves in a convolution model for phase transitions (1997) Arch. Rat. Mech. Anal, 138, pp. 105-136
  • Carr, J., Chmaj, A., Uniqueness of traveling waves for nonlocal monostable equations (2004) Proc. Amer. Math. Soc, 132, pp. 2433-2439
  • Chasseigne, E., Chaves, M., Rossi, J.D., Asymptotic behavior for nonlocal diffusion equations (2006) J. Math. Pures Appl, 86, pp. 271-291
  • Chen, X., Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations (1997) Adv. Diff. Eqns, 2, pp. 125-160
  • Chmaj, A., Ren, X., Homoclinic solutions of an integral equation: Existence and stability (1999) J. Diff. Eqns, 155, pp. 17-43
  • Chmaj, A., Ren, X., The nonlocal bistable equation: Stationary solutions on a bounded interval (2002) Electronic J. Diff. Eqns, 2002, pp. 1-12
  • Cortázar, C., Coville, J., Elgueta, M., Martínez, S., A non local inhomogeneous dispersal process (2007) J. Diff. Eqns, 241, pp. 332-358
  • Cortázar, C., Elgueta, M., Rossi, J.D., A nonlocal diffusion equation whose solutions develop a free boundary (2005) Ann. Henri Poincaré, 6, pp. 269-281
  • Cortázar, C., Elgueta, M., Rossi, J.D., Wolanski, N., Boundary fluxes for nonlocal diffusion (2007) J. Diff. Eqns, 234, pp. 360-390
  • Cortázar, 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, pp. 137-156
  • 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 (2007) Electron. J. Diff. Eqns, 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, pp. 1693-1709
  • Coville, J., Dupaigne, L., Propagation speed of travelling fronts in nonlocal reaction diffusion equations (2005) Nonl. 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
  • del Pino, M.A., Positive solutions of a semilinear elliptic equation on a compact manifold (1994) Nonlinear Anal, 22, pp. 1423-1430
  • Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions (2003) Trends in Nonlinear Analysis, pp. 153-191. , pp, Springer-Verlag, Berlin
  • Fraile, J.M., Koch-Medina, P., Lopez-Gomez, J., Merino, S., Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear equation (1996) J. Diff. Eqns, 127, pp. 295-319
  • García-Melián, J., Gomez-Reñasco, R., Lopez-Gomez, J., Sabina de Lis, J., Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs (1998) Arch. Rat. Mech. Anal, 145, pp. 261-289
  • García-Melián, J., Letelier-Albornoz, R., Sabina de Lis, J., Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up (2001) Proc. Amer. Math. Soc, 129, pp. 3593-3602
  • García-Melián, J., Rossi, J.D., On the principal eigenvalue of some nonlocal diffusion problems (2009) J. Diff. Eqns, 246, pp. 21-38
  • García-Melián, J., Sabina de Lis, J., On the perturbation of eigenvalues for the p-Laplacian (2001) C. R. Acad. Sci. Paris Sér. I Math, 332, pp. 893-898
  • Hutson, V., Martínez, S., Mischaikow, K., Vickers, G.T., The evolution of dispersal (2003) J. Math. Biol, 47, pp. 483-517
  • Ignat, L.I., Rossi, J.D., Refined asymptotic expansions for nonlocal evolution equations (2008) J. Evol. Eqns, 8, pp. 617-629
  • Ignat, L.I., Rossi, J.D., A nonlocal convection-diffusion equation (2007) J. Funct. Anal, 251, pp. 399-437
  • Krein, M.G., Rutman, M.A., Linear operators leaving invariant a cone in a Banach space (1962) Amer. Math. Soc. Transl, 10, pp. 199-325
  • Lopez-Gomez, J., Sabina de Lis, J., First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs (1998) J. Diff. Eqns, 148, pp. 47-64
  • Ouyang, T., On the positive solutions of semilinear equations Δu + λu - hup = 0 on the compact manifolds (1992) Trans. Amer. Math. Soc, 331, pp. 503-527
  • Pazoto, A.F., Rossi, J.D., Asymptotic behavior for a semilinear nonlocal equation (2007) Asympt. Anal, 52, pp. 143-155
  • Schumacher, K., Travelling-front solutions for integro-differential equations I (1980) J. Reine Angew. Math, 316, pp. 54-70
  • Zhang, L., Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks (2004) J. Diff. Eqns, 197, pp. 162-196

Citas:

---------- APA ----------
García-Melián, J. & Rossi, J.D. (2009) . A logistic equation with refuge and nonlocal diffusion. Communications on Pure and Applied Analysis, 8(6), 2037-2053.
http://dx.doi.org/10.3934/cpaa.2009.8.2037
---------- CHICAGO ----------
García-Melián, J., Rossi, J.D. "A logistic equation with refuge and nonlocal diffusion" . Communications on Pure and Applied Analysis 8, no. 6 (2009) : 2037-2053.
http://dx.doi.org/10.3934/cpaa.2009.8.2037
---------- MLA ----------
García-Melián, J., Rossi, J.D. "A logistic equation with refuge and nonlocal diffusion" . Communications on Pure and Applied Analysis, vol. 8, no. 6, 2009, pp. 2037-2053.
http://dx.doi.org/10.3934/cpaa.2009.8.2037
---------- VANCOUVER ----------
García-Melián, J., Rossi, J.D. A logistic equation with refuge and nonlocal diffusion. Commun. Pure Appl. Anal. 2009;8(6):2037-2053.
http://dx.doi.org/10.3934/cpaa.2009.8.2037