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Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces
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 |
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.
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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