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doi: 10.3934/dcds.2021170
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[3]

R. G. Bartle, The Elements of Real Analysis, 2$^{ed}$ edition, John Wiley & Sons, New York-London-Sydney, 1976.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6] D. ben Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511605826.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956.   Google Scholar
[17]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.  Google Scholar

[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.  Google Scholar

[19] J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511470943.  Google Scholar
[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.  Google Scholar

[21]

J. Murray,, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[23]

A. Rodríguez-Bernal,, Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces, Submitted, 2020. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] R. S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. G. Bartle, The Elements of Real Analysis, 2$^{ed}$ edition, John Wiley & Sons, New York-London-Sydney, 1976.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6] D. ben Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511605826.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956.   Google Scholar
[17]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.  Google Scholar

[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.  Google Scholar

[19] J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511470943.  Google Scholar
[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.  Google Scholar

[21]

J. Murray,, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[23]

A. Rodríguez-Bernal,, Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces, Submitted, 2020. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] R. S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006.   Google Scholar
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