Monotone systems involving variable-order nonlocal operators
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Miguel Yangari
Escuela Politécnica Nacional,Ecuador. Departamento de Matemática
In this paper, we study the existence and uniqueness of bounded viscosity solutions for parabolic Hamilton–Jacobi monotone systems in which the diffusion term is driven by variable-order nonlocal operators whose kernels depend on the space-time variable. We prove the existence of solutions via Perron’s method, and considering Hamiltonians with linear and superlinear nonlinearities related to their gradient growth we state a comparison principle for bounded sub and supersolutions.
Moreover, we present steady-state large time behavior with an exponential rate of convergence.
Paraules clau
viscosity solutions, Hamilton–Jacobi, variable-order nonlocal operators, comparison principles, large time behavior
Article Details
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Yangari, Miguel. «Monotone systems involving variable-order nonlocal operators». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 1, p. 129-58, http://raco.cat/index.php/PublicacionsMatematiques/article/view/396418.
Referències
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equations, Ann. Inst. H. Poincar´e Anal. Non Linéaire 13(3) (1996), 293–317.
DOI: 10.1016/S0294-1449(16)30106-8.
M. Arisawa, Homogenization of a class of integro-differential equations with
Lévy operators, Comm. Partial Differential Equations 34(7) (2009), 617–624.
DOI: 10.1080/03605300902963518.
J. Bae, Regularity for fully nonlinear equations driven by spatial-inhomogeneous
nonlocal operators, Potential Anal. 43(4) (2015), 611–624. DOI: 10.1007/
s11118-015-9488-z.
J. Bae and M. Kassmann, Schauder estimates in generalized Hölder spaces,
Preprint (2015). arXiv:1505.05498.
G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential
equations and integral-partial differential equations, Stochastics Stochastics Rep.
60(1–2) (1997), 57–83. DOI: 10.1080/17442509708834099.
G. Barles and C. Imbert, Second-order elliptic integro-differential equations:
viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire
25(3) (2008), 567–585. DOI: 10.1016/j.anihpc.2007.02.007.
G. Barles, S. Koike, O. Ley, and E. Topp, Regularity results and large
time behavior for integro-differential equations with coercive Hamiltonians,
Calc. Var. Partial Differential Equations 54(1) (2015), 539–572. DOI: 10.1007/
s00526-014-0794-x.
G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim. 26(5) (1988), 1133–1148.
DOI: 10.1137/0326063.
R. F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory
Related Fields 79(2) (1988), 271–287. DOI: 10.1007/BF00320922.
R. F. Bass, Occupation time densities for stable-like processes and other pure
jump Markov processes, Stochastic Process. Appl. 29(1) (1988), 65–83. DOI: 10.
1016/0304-4149(88)90028-2.
R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal. 257(8)
(2009), 2693–2722. DOI: 10.1016/j.jfa.2009.05.012.
R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with
respect to operators of variable order, Comm. Partial Differential Equations
30(8) (2005), 1249–1259. DOI: 10.1080/03605300500257677.
R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of
variable order, Trans. Amer. Math. Soc. 357(2) (2005), 837–850. DOI: 10.1090/
S0002-9947-04-03549-4.
R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential
Anal. 17(4) (2002), 375–388. DOI: 10.1023/A:1016378210944.
F. E. Benth, K. H. Karlsen, and K. Reikvam, Optimal portfolio selection
with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance Stoch. 5(3) (2001), 275–303.
DOI: 10.1007/PL00013538.
J. Bertoin, “Lévy Processes”, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996.
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Comm. Pure Appl. Math. 62(5) (2009), 597–638. DOI: 10.
1002/cpa.20274.
F. Camilli, O. Ley, and P. Loreti, Homogenization of monotone systems of
Hamilton–Jacobi equations, ESAIM Control Optim. Calc. Var. 16(1) (2010),
58–76. DOI: 10.1051/cocv:2008061.
P. Cardaliaguet, Solutions de viscosité d’équations elliptiques et paraboliques
non linéaires, Notes d’un cours de DEA, Université de Rennes (2004).
X. Chen, Z.-Q. Chen, and J. Wang, Heat kernel for non-local operators with
variable order, Stochastic Process. Appl. 130(6) (2020), 3574–3647. DOI: 10.
1016/j.spa.2019.10.004.
Z.-Q. Chen and J.-M. Wang, Perturbation by non-local operators, Ann. Inst.
Henri Poincaré Probab. Stat. 54(2) (2018), 606–639. DOI: 10.1214/16-AIHP816.
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277(1) (1983), 1–42. DOI: 10.2307/1999343.
H. Engler and S. M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton–Jacobi equations, Proc. London Math. Soc. (3) 63(1) (1991),
212–240. DOI: 10.1112/plms/s3-63.1.212.
H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55(2)
(1987), 369–384. DOI: 10.1215/S0012-7094-87-05521-9.
H. Ishii and S. Koike, Viscosity solutions for monotone systems of secondorder elliptic PDEs, Comm. Partial Differential Equations 16(6–7) (1991),
1095–1128. DOI: 10.1080/03605309108820791.
N. Jacob, Further pseudodifferential operators generating Feller semigroups and
Dirichlet forms, Rev. Mat. Iberoamericana 9(2) (1993), 373–407. DOI: 10.4171/
RMI/141.
N. Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z. 215(1) (1994), 151–166. DOI: 10.1007/BF02571704.
N. Jacob and H.-G. Leopold, Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integral Equations Operator
Theory 17(4) (1993), 544–553. DOI: 10.1007/BF01200393.
E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for
viscosity solutions of integro-PDEs, J. Differential Equations 212(2) (2005),
278–318. DOI: 10.1016/j.jde.2004.06.021.
E. R. Jakobsen and K. H. Karlsen, A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA
Nonlinear Differential Equations Appl. 13(2) (2006), 137–165. DOI: 10.1007/
s00030-005-0031-6.
K. Kikuchi and A. Negoro, On Markov process generated by pseudodifferential
operator of variable order, Osaka J. Math. 34(2) (1997), 319–335. DOI: 10.
18910/12776.
Y.-C. Kim and K.-A. Lee, Regularity results for fully nonlinear integrodifferential operators with nonsymmetric positive kernels, Manuscripta Math.
139(3–4) (2012), 291–319. DOI: 10.1007/s00229-011-0516-z.
Y.-C. Kim and K.-A. Lee, Regularity results for fully nonlinear integrodifferential operators with nonsymmetric positive kernels: subcritical case, Potential Anal. 38(2) (2013), 433–455. DOI: 10.1007/s11118-012-9280-2.
S. Koike, Uniqueness of viscosity solutions for monotone systems of fully nonlinear PDEs under Dirichlet condition, Nonlinear Anal. 22(4) (1994), 519–532.
DOI: 10.1016/0362-546X(94)90172-4.
H. G. Leopold of Jena, On a class of function spaces and related pseudodifferential operators, Math. Nachr. 127(1) (1986), 65–82. DOI: 10.1002/mana.
19861270106.
H.-G. Leopold, On Besov spaces of variable order of differentiation, Z. Anal.
Anwendungen 8(1) (1989), 69–82. DOI: 10.4171/ZAA/337.
H.-G. Leopold, On function spaces of variable order of differentiation, Forum
Math. 3(1) (1991), 1–21. DOI: 10.1515/form.1991.3.1.
O. Ley and V. D. Nguyen, Gradient bounds for nonlinear degenerate parabolic
equations and application to large time behavior of systems, Nonlinear Anal. 130
(2016), 76–101. DOI: 10.1016/j.na.2015.09.012.
D. Luo and J. Wang, Coupling by reflection and Hölder regularity for non-local
operators of variable order, Trans. Amer. Math. Soc. 371(1) (2019), 431–459.
DOI: 10.1090/tran/7259.
H. Mitake and H. V. Tran, A dynamical approach to the large-time behavior
of solutions to weakly coupled systems of Hamilton–Jacobi equations, J. Math.
Pures Appl. (9) 101(1) (2014), 76–93. DOI: 10.1016/j.matpur.2013.05.004.
A. Negoro, Stable-like processes: construction of the transition density and the
behavior of sample paths near t = 0, Osaka J. Math. 31(1) (1994), 189–214.
DOI: 10.18910/11079.
B. Øksendal and A. Sulem, “Applied Stochastic Control of Jump Diffusions”,
Universitext, Springer-Verlag, Berlin, 2005. DOI: 10.1007/b137590.
H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity
solution approach, J. Math. Systems Estim. Control 8(1) (1998), 27 pp.
K.-I. Sato, “Lévy Processes and Infinitely Divisible Distributions”, Translated
from the 1990 Japanese original, Revised by the author, Cambridge Studies in
Advanced Mathematics 68, Cambridge University Press, Cambridge, 1999.
A. Sayah, Equations d’Hamilton–Jacobi du premier ordre avec termes intégrodifférentiels. Partie I: Unicité des solutions de viscosité, Comm. Partial Differential Equations 16(6–7) (1991), 1057–1074. DOI: 10.1080/03605309108820789.
Partie II: Existence de solutions de viscosité, Comm. Partial Differential Equations 16(6–7) (1991), 1075–1093. DOI: 10.1080/03605309108820790.
L. Silvestre, Hölder estimates for solutions of integro-differential equations
like the fractional Laplace, Indiana Univ. Math. J. 55(3) (2006), 1155–1174.
DOI: 10.1512/iumj.2006.55.2706.
H. M. Soner, Optimal control with state-space constraint. II, SIAM J. Control
Optim. 24(6) (1986), 1110–1122. DOI: 10.1137/0324067.
E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with
Caputo time derivative, J. Differential Equations 262(12) (2017), 6018–6046.
DOI: 10.1016/j.jde.2017.02.024.
M. Tsuchiya, Lévy measure with generalized polar decomposition and the associated SDE with jumps, Stochastics Stochastics Rep. 38(2) (1992), 95–117.
DOI: 10.1080/17442509208833748.
J.-M. Wang, Laplacian perturbed by non-local operators, Math. Z. 279(1–2)
(2015), 521–556. DOI: 10.1007/s00209-014-1380-9.
L. Wang and B. Zhang, Infinitely many solutions for Kirchhoff-type variableorder fractional Laplacian problems involving variable exponents, Appl. Anal.
(2019), pp. 1–18. DOI: 10.1080/00036811.2019.1688790.
W. A. Woyczynskí , Lévy processes in the physical sciences, in: “Lévy Processes”, Birkh¨auser Boston, Boston, MA, 2001, pp. 241–266.
M. Xiang, B. Zhang, and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019),
190–204. DOI: 10.1016/j.na.2018.07.016.
equations, Ann. Inst. H. Poincar´e Anal. Non Linéaire 13(3) (1996), 293–317.
DOI: 10.1016/S0294-1449(16)30106-8.
M. Arisawa, Homogenization of a class of integro-differential equations with
Lévy operators, Comm. Partial Differential Equations 34(7) (2009), 617–624.
DOI: 10.1080/03605300902963518.
J. Bae, Regularity for fully nonlinear equations driven by spatial-inhomogeneous
nonlocal operators, Potential Anal. 43(4) (2015), 611–624. DOI: 10.1007/
s11118-015-9488-z.
J. Bae and M. Kassmann, Schauder estimates in generalized Hölder spaces,
Preprint (2015). arXiv:1505.05498.
G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential
equations and integral-partial differential equations, Stochastics Stochastics Rep.
60(1–2) (1997), 57–83. DOI: 10.1080/17442509708834099.
G. Barles and C. Imbert, Second-order elliptic integro-differential equations:
viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire
25(3) (2008), 567–585. DOI: 10.1016/j.anihpc.2007.02.007.
G. Barles, S. Koike, O. Ley, and E. Topp, Regularity results and large
time behavior for integro-differential equations with coercive Hamiltonians,
Calc. Var. Partial Differential Equations 54(1) (2015), 539–572. DOI: 10.1007/
s00526-014-0794-x.
G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim. 26(5) (1988), 1133–1148.
DOI: 10.1137/0326063.
R. F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory
Related Fields 79(2) (1988), 271–287. DOI: 10.1007/BF00320922.
R. F. Bass, Occupation time densities for stable-like processes and other pure
jump Markov processes, Stochastic Process. Appl. 29(1) (1988), 65–83. DOI: 10.
1016/0304-4149(88)90028-2.
R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal. 257(8)
(2009), 2693–2722. DOI: 10.1016/j.jfa.2009.05.012.
R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with
respect to operators of variable order, Comm. Partial Differential Equations
30(8) (2005), 1249–1259. DOI: 10.1080/03605300500257677.
R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of
variable order, Trans. Amer. Math. Soc. 357(2) (2005), 837–850. DOI: 10.1090/
S0002-9947-04-03549-4.
R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential
Anal. 17(4) (2002), 375–388. DOI: 10.1023/A:1016378210944.
F. E. Benth, K. H. Karlsen, and K. Reikvam, Optimal portfolio selection
with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance Stoch. 5(3) (2001), 275–303.
DOI: 10.1007/PL00013538.
J. Bertoin, “Lévy Processes”, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996.
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Comm. Pure Appl. Math. 62(5) (2009), 597–638. DOI: 10.
1002/cpa.20274.
F. Camilli, O. Ley, and P. Loreti, Homogenization of monotone systems of
Hamilton–Jacobi equations, ESAIM Control Optim. Calc. Var. 16(1) (2010),
58–76. DOI: 10.1051/cocv:2008061.
P. Cardaliaguet, Solutions de viscosité d’équations elliptiques et paraboliques
non linéaires, Notes d’un cours de DEA, Université de Rennes (2004).
X. Chen, Z.-Q. Chen, and J. Wang, Heat kernel for non-local operators with
variable order, Stochastic Process. Appl. 130(6) (2020), 3574–3647. DOI: 10.
1016/j.spa.2019.10.004.
Z.-Q. Chen and J.-M. Wang, Perturbation by non-local operators, Ann. Inst.
Henri Poincaré Probab. Stat. 54(2) (2018), 606–639. DOI: 10.1214/16-AIHP816.
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277(1) (1983), 1–42. DOI: 10.2307/1999343.
H. Engler and S. M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton–Jacobi equations, Proc. London Math. Soc. (3) 63(1) (1991),
212–240. DOI: 10.1112/plms/s3-63.1.212.
H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55(2)
(1987), 369–384. DOI: 10.1215/S0012-7094-87-05521-9.
H. Ishii and S. Koike, Viscosity solutions for monotone systems of secondorder elliptic PDEs, Comm. Partial Differential Equations 16(6–7) (1991),
1095–1128. DOI: 10.1080/03605309108820791.
N. Jacob, Further pseudodifferential operators generating Feller semigroups and
Dirichlet forms, Rev. Mat. Iberoamericana 9(2) (1993), 373–407. DOI: 10.4171/
RMI/141.
N. Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z. 215(1) (1994), 151–166. DOI: 10.1007/BF02571704.
N. Jacob and H.-G. Leopold, Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integral Equations Operator
Theory 17(4) (1993), 544–553. DOI: 10.1007/BF01200393.
E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for
viscosity solutions of integro-PDEs, J. Differential Equations 212(2) (2005),
278–318. DOI: 10.1016/j.jde.2004.06.021.
E. R. Jakobsen and K. H. Karlsen, A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA
Nonlinear Differential Equations Appl. 13(2) (2006), 137–165. DOI: 10.1007/
s00030-005-0031-6.
K. Kikuchi and A. Negoro, On Markov process generated by pseudodifferential
operator of variable order, Osaka J. Math. 34(2) (1997), 319–335. DOI: 10.
18910/12776.
Y.-C. Kim and K.-A. Lee, Regularity results for fully nonlinear integrodifferential operators with nonsymmetric positive kernels, Manuscripta Math.
139(3–4) (2012), 291–319. DOI: 10.1007/s00229-011-0516-z.
Y.-C. Kim and K.-A. Lee, Regularity results for fully nonlinear integrodifferential operators with nonsymmetric positive kernels: subcritical case, Potential Anal. 38(2) (2013), 433–455. DOI: 10.1007/s11118-012-9280-2.
S. Koike, Uniqueness of viscosity solutions for monotone systems of fully nonlinear PDEs under Dirichlet condition, Nonlinear Anal. 22(4) (1994), 519–532.
DOI: 10.1016/0362-546X(94)90172-4.
H. G. Leopold of Jena, On a class of function spaces and related pseudodifferential operators, Math. Nachr. 127(1) (1986), 65–82. DOI: 10.1002/mana.
19861270106.
H.-G. Leopold, On Besov spaces of variable order of differentiation, Z. Anal.
Anwendungen 8(1) (1989), 69–82. DOI: 10.4171/ZAA/337.
H.-G. Leopold, On function spaces of variable order of differentiation, Forum
Math. 3(1) (1991), 1–21. DOI: 10.1515/form.1991.3.1.
O. Ley and V. D. Nguyen, Gradient bounds for nonlinear degenerate parabolic
equations and application to large time behavior of systems, Nonlinear Anal. 130
(2016), 76–101. DOI: 10.1016/j.na.2015.09.012.
D. Luo and J. Wang, Coupling by reflection and Hölder regularity for non-local
operators of variable order, Trans. Amer. Math. Soc. 371(1) (2019), 431–459.
DOI: 10.1090/tran/7259.
H. Mitake and H. V. Tran, A dynamical approach to the large-time behavior
of solutions to weakly coupled systems of Hamilton–Jacobi equations, J. Math.
Pures Appl. (9) 101(1) (2014), 76–93. DOI: 10.1016/j.matpur.2013.05.004.
A. Negoro, Stable-like processes: construction of the transition density and the
behavior of sample paths near t = 0, Osaka J. Math. 31(1) (1994), 189–214.
DOI: 10.18910/11079.
B. Øksendal and A. Sulem, “Applied Stochastic Control of Jump Diffusions”,
Universitext, Springer-Verlag, Berlin, 2005. DOI: 10.1007/b137590.
H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity
solution approach, J. Math. Systems Estim. Control 8(1) (1998), 27 pp.
K.-I. Sato, “Lévy Processes and Infinitely Divisible Distributions”, Translated
from the 1990 Japanese original, Revised by the author, Cambridge Studies in
Advanced Mathematics 68, Cambridge University Press, Cambridge, 1999.
A. Sayah, Equations d’Hamilton–Jacobi du premier ordre avec termes intégrodifférentiels. Partie I: Unicité des solutions de viscosité, Comm. Partial Differential Equations 16(6–7) (1991), 1057–1074. DOI: 10.1080/03605309108820789.
Partie II: Existence de solutions de viscosité, Comm. Partial Differential Equations 16(6–7) (1991), 1075–1093. DOI: 10.1080/03605309108820790.
L. Silvestre, Hölder estimates for solutions of integro-differential equations
like the fractional Laplace, Indiana Univ. Math. J. 55(3) (2006), 1155–1174.
DOI: 10.1512/iumj.2006.55.2706.
H. M. Soner, Optimal control with state-space constraint. II, SIAM J. Control
Optim. 24(6) (1986), 1110–1122. DOI: 10.1137/0324067.
E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with
Caputo time derivative, J. Differential Equations 262(12) (2017), 6018–6046.
DOI: 10.1016/j.jde.2017.02.024.
M. Tsuchiya, Lévy measure with generalized polar decomposition and the associated SDE with jumps, Stochastics Stochastics Rep. 38(2) (1992), 95–117.
DOI: 10.1080/17442509208833748.
J.-M. Wang, Laplacian perturbed by non-local operators, Math. Z. 279(1–2)
(2015), 521–556. DOI: 10.1007/s00209-014-1380-9.
L. Wang and B. Zhang, Infinitely many solutions for Kirchhoff-type variableorder fractional Laplacian problems involving variable exponents, Appl. Anal.
(2019), pp. 1–18. DOI: 10.1080/00036811.2019.1688790.
W. A. Woyczynskí , Lévy processes in the physical sciences, in: “Lévy Processes”, Birkh¨auser Boston, Boston, MA, 2001, pp. 241–266.
M. Xiang, B. Zhang, and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019),
190–204. DOI: 10.1016/j.na.2018.07.016.