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

Fundamental solutions and decay of fully non-local problems

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.

Citation: Juan C. Pozo, Vicente Vergara. Fundamental solutions and decay of fully non-local problems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 639-666. doi: 10.3934/dcds.2019026
References:
[1]

B. n. BarriosI. PeralF. 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.

[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.

[3]

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

[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.

[5]

L. A. CaffarelliJ.-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.

[6]

R. CarloneA. 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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

C. ImbertR. 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.

[17]

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

[18]

J. KemppainenJ. SiljanderV. 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.

[19]

J. KemppainenJ. 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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

show all references

References:
[1]

B. n. BarriosI. PeralF. 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.

[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.

[3]

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

[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.

[5]

L. A. CaffarelliJ.-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.

[6]

R. CarloneA. 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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

C. ImbertR. 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.

[17]

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

[18]

J. KemppainenJ. SiljanderV. 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.

[19]

J. KemppainenJ. 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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

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