Advanced Search
Article Contents
Article Contents

Nonlinear nonlocal reaction-diffusion problem with local reaction

  • * Corresponding author: Aníbal Rodríguez-Bernal

    * Corresponding author: Aníbal Rodríguez-Bernal 

The first author is supported by the projects Projects MTM2016-75465 and PID2019-103860GB-I00, MINECO, Spain and Grupo CADEDIF GR58/08, Grupo 920894. Second author is partially supported by projects MTM2016-75465, MTM2017-83391 and Grupo CADEDIF GR58/08, Grupo 920894, Junta de Andalucía FQM-131. ICMAT is partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO)

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.

    Mathematics Subject Classification: Primary: 37L15, 45G, 45M05, 45M10, 45M20, 45P05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 2010. doi: 10.1090/surv/165.
    [2] J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential Equations, 25 (2000), 1-37.  doi: 10.1080/03605300008821506.
    [3] R. G. Bartle, The Elements of Real Analysis, 2$^{ed}$ edition, John Wiley & Sons, New York-London-Sydney, 1976.
    [4] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Statist. Phys., 95 (1999), 1119-1139. 
    [5] P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.
    [6] D. ben Avraham and  S. HavlinDiffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511605826.
    [7] H. BerestyckiJ. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.
    [8] H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.
    [9] E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.
    [10] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. 
    [11] E. ChristensenC. Ivan and M. L. Lapidus, Dirac operators and spectral triples for some fractal sets built on curves, Adv. Math., 217 (2008), 42-78.  doi: 10.1016/j.aim.2007.06.009.
    [12] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.
    [13] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.
    [14] J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.
    [15] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.
    [16] J. CrankThe Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956. 
    [17] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.
    [18] V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.
    [19] J. KigamiAnalysis on Fractals, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511470943.
    [20] J. Kigami, Measurable Riemannian geometry on the Sierpinski gasket: The Kusuoka measure and the Gaussian heat kernel estimate, Math. Ann., 340 (2008), 781-804.  doi: 10.1007/s00208-007-0169-0.
    [21] J. Murray,, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.
    [22] A. Pazy,, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
    [23] A. Rodríguez-Bernal,, Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces, Submitted, 2020.
    [24] A. Rodríguez-Bernal and S. Sastre-Gómez, Linear non-local diffusion problems in metric measure spaces, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 833-863.  doi: 10.1017/S0308210515000724.
    [25] A. Rodríguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications, J. Differential Equations, 244 (2008), 2983-3030.  doi: 10.1016/j.jde.2008.02.046.
    [26] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.
    [27] W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.  doi: 10.3934/dcds.2015.35.1665.
    [28] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.
    [29] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.
    [30] R. S. StrichartzDifferential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006. 
  • 加载中

Article Metrics

HTML views(1774) PDF downloads(324) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint