American Institute of Mathematical Sciences

Advanced
January  2019, 39(1): 639-666. doi: 10.3934/dcds.2019026

Fundamental solutions and decay of fully non-local problems

Juan C. Pozo 1,*,, and  Vicente Vergara 2,†,

1. 

Departamento de Matemáticas y Estadísticas, Facultad de Ingeniería y Ciencias, Universidad de La Frontera, Temuco, Chile
2. 

Departamento de Matemáticas, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile

‡ Corresponding author: Juan C. Pozo

*The first author is partially supported by Fondecyt grant 11160295.
†The second author is partially supported by Fondecyt grant 1150230.

Received  June 2018 Revised  July 2018 Published  October 2018

In this paper, we study a fully non-local reaction-diffusion equation which is non-local both in time and space. We apply subordination principles to construct the fundamental solutions of this problem, which we use to find a representation of the mild solutions. Moreover, using techniques of Harmonic Analysis and Fourier Multipliers, we obtain the temporal decay rates for the mild solutions.

Keywords: Non-local partial differential equations, large-time behavior of solutions, fundamental solutions and subordination methods.
Mathematics Subject Classification: Primary: 35B40, 35R11, 35E05; Secondary: 42A45.
Citation: Juan C. Pozo, Vicente Vergara. Fundamental solutions and decay of fully non-local problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 639-666. doi: 10.3934/dcds.2019026
References:
[1]

B. n. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1.  Google Scholar

[2]

S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368-370.  doi: 10.1073/pnas.35.7.368.  Google Scholar

[3]

S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.  Google Scholar

[4]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.  Google Scholar

[5]

L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[6]

R. Carlone, A. Fiorenza and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273 (2017), 1258-1294.  doi: 10.1016/j.jfa.2017.04.013.  Google Scholar

[7]

P. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10 (1979), 365-388.  doi: 10.1137/0510035.  Google Scholar

[8]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.  Google Scholar

[9]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[10]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.  Google Scholar

[11]

G. B. Folland, Real Analysis, 2nd edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.  Google Scholar

[12]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[13]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, vol. 34 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[14]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl. (9), 92 (2009), 163-187.  doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[15]

C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/ε)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89.  doi: 10.1007/s00205-007-0074-4.  Google Scholar

[16]

C. Imbert, R. Monneau and E. Rouy, Homogenization of first order equations with $(u/ε)$-periodic Hamiltonians. II. Application to dislocations dynamics, Comm. Partial Differential Equations, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[17]

N. Jacob, Pseudo-differential Operators and Markov Processes, vol. 94 of Mathematical Research, Akademie Verlag, Berlin, 1996.  Google Scholar

[18]

J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar

[19]

J. Kemppainen, J. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[20]

K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[21]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.  Google Scholar

[22]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993, [2012] reprint of the 1993 edition. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[25]

K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999, Translated from the 1990 Japanese original, Revised by the author.  Google Scholar

[26]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, vol. 37 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2010, Theory and applications.  Google Scholar

[27]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[28]

J. L. Vázquez, Asymptotic behaviour for the fractional heat equation in the Euclidean space, Complex Var. Elliptic Equ., 63 (2018), 1216-1231.  doi: 10.1080/17476933.2017.1393807.  Google Scholar

[29]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.  Google Scholar

[30]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[31]

R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.  doi: 10.1016/j.jmaa.2008.06.054.  Google Scholar

show all references

References:
[1]

B. n. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1.  Google Scholar

[2]

S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368-370.  doi: 10.1073/pnas.35.7.368.  Google Scholar

[3]

S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.  Google Scholar

[4]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.  Google Scholar

[5]

L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[6]

R. Carlone, A. Fiorenza and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273 (2017), 1258-1294.  doi: 10.1016/j.jfa.2017.04.013.  Google Scholar

[7]

P. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10 (1979), 365-388.  doi: 10.1137/0510035.  Google Scholar

[8]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.  Google Scholar

[9]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[10]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.  Google Scholar

[11]

G. B. Folland, Real Analysis, 2nd edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.  Google Scholar

[12]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[13]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, vol. 34 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[14]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl. (9), 92 (2009), 163-187.  doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[15]

C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/ε)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89.  doi: 10.1007/s00205-007-0074-4.  Google Scholar

[16]

C. Imbert, R. Monneau and E. Rouy, Homogenization of first order equations with $(u/ε)$-periodic Hamiltonians. II. Application to dislocations dynamics, Comm. Partial Differential Equations, 33 (2008), 479-516.  doi: 10.1080/03605300701318922.  Google Scholar

[17]

N. Jacob, Pseudo-differential Operators and Markov Processes, vol. 94 of Mathematical Research, Akademie Verlag, Berlin, 1996.  Google Scholar

[18]

J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar

[19]

J. Kemppainen, J. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[20]

K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[21]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.  Google Scholar

[22]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993, [2012] reprint of the 1993 edition. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[25]

K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999, Translated from the 1990 Japanese original, Revised by the author.  Google Scholar

[26]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, vol. 37 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2010, Theory and applications.  Google Scholar

[27]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar

[28]

J. L. Vázquez, Asymptotic behaviour for the fractional heat equation in the Euclidean space, Complex Var. Elliptic Equ., 63 (2018), 1216-1231.  doi: 10.1080/17476933.2017.1393807.  Google Scholar

[29]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.  Google Scholar

[30]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[31]

R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.  doi: 10.1016/j.jmaa.2008.06.054.  Google Scholar

[1]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[2]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
[3]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[4]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096
[5]

Michael Herty, Reinhard Illner. Coupling of non-local driving behaviour with fundamental diagrams. Kinetic & Related Models, 2012, 5 (4) : 843-855. doi: 10.3934/krm.2012.5.843
[6]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[7]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229
[8]

Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
[9]

Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055
[10]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053
[11]

Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653
[12]

Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132
[13]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[14]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[15]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020137
[16]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
[17]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041
[18]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[19]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[20]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (123)
  • HTML views (93)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences