Fundamental solutions and decay of fully non-local problems

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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 and Continuous Dynamical Systems, 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.
[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. 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.
[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.
[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. 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.
[17]	N. Jacob, Pseudo-differential Operators and Markov Processes, vol. 94 of Mathematical Research, Akademie Verlag, Berlin, 1996.
[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.
[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.
[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. 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.
[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. 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.
[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.
[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. 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.
[17]	N. Jacob, Pseudo-differential Operators and Markov Processes, vol. 94 of Mathematical Research, Akademie Verlag, Berlin, 1996.
[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.
[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.
[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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