April  2022, 42(4): 1731-1765. doi: 10.3934/dcds.2021170

Nonlinear nonlocal reaction-diffusion problem with local reaction

1. 

Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain, and, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41012, Sevilla, Spain

* Corresponding author: Aníbal Rodríguez-Bernal

Received  March 2021 Revised  August 2021 Published  April 2022 Early access  November 2021

Fund Project: 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)

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.

Citation: Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170
References:
[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. Havlin, Diffusion 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. Crank, The 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. Kigami, Analysis 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. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006. 

show all references

References:
[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. Havlin, Diffusion 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. Crank, The 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. Kigami, Analysis 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. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006. 
[1]

Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200

[2]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure and Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037

[3]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311

[4]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[5]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations and Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[6]

Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005

[7]

Pengchao Lai, Qi Li. Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3313-3323. doi: 10.3934/dcdsb.2021186

[8]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471

[9]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343

[10]

Armel Ovono Andami. From local to nonlocal in a diffusion model. Conference Publications, 2011, 2011 (Special) : 54-60. doi: 10.3934/proc.2011.2011.54

[11]

Xavier Cabré, Manel Sanchón. Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Communications on Pure and Applied Analysis, 2007, 6 (1) : 43-67. doi: 10.3934/cpaa.2007.6.43

[12]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

[13]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[14]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157

[15]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681

[16]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65

[17]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775

[18]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

[19]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[20]

Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (270)
  • HTML views (189)
  • Cited by (0)

[Back to Top]