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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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