October  2018, 38(10): 5221-5243. doi: 10.3934/dcds.2018231

Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile

* Corresponding author: Andrei Rodríguez

Received  March 2018 Revised  May 2018 Published  July 2018

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are a) the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all Hölder-continuous initial data, with Hölder exponent above a critical a value, and b) the nonexistence of solutions satisfying the boundary data in the classical sense for all time. In this case, the phenomenon of loss of boundary conditions occurs in finite time, depending on a largeness condition on the initial data.

Citation: Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231
References:
[1]

A. Attouchi, Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion, Journal of Differential Equations, 253 (2012), 2474-2492. doi: 10.1016/j.jde.2012.07.002.

[2]

A. Attouchi and P. Souplet, Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with p-Laplacian diffusion, Transactions of the American Mathematical Society, 369 (2017), 935-974. doi: 10.1090/tran/6684.

[3]

G. BarlesE. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana University Mathematics Journal, 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315.

[4]

G. Barles and F. Da Lio, On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations, Journal de Mathématiques Pures et Appliquées, 83 (2004), 53-75. doi: 10.1016/S0021-7824(03)00070-9.

[5]

G. BarlesS. KoikeO. Ley and E. Topp, Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians, Calculus of Variations and Partial Differential Equations, 54 (2015), 539-572. doi: 10.1007/s00526-014-0794-x.

[6]

G. BarlesO. Ley and E. Topp, Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior, Nonlinearity, 30 (2017), 703-734. doi: 10.1088/1361-6544/aa527f.

[7]

G. Barles and E. Topp, Existence, uniqueness, and asymptotic behavior for nonlocal parabolic problems with dominating gradient terms, SIAM Journal on Mathematical Analysis, 48 (2016), 1512-1547. doi: 10.1137/140967192.

[8]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.

[9]

M. Crandall, M. Kocan, P. Soravia and A. Swiech, On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, in Progress in Elliptic and Parabolic Partial Differential Equations, vol. 350, Pitman Research Notes in Mathematics Series, 1996,136-162.

[10]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[11]

G. DávilaA. Quaas and E. Topp, Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms, Mathematische Annalen, 369 (2017), 1211-1236. doi: 10.1007/s00208-016-1481-3.

[12]

L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, Journal of Differential Equations, 263 (2017), 765-778. doi: 10.1016/j.jde.2017.02.051.

[13]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[14]

F. Dragoni, Introduction to Viscosity Solutions for Nonlinear PDEs, Available at https://orca-mwe.cf.ac.uk/75040/1/ViscSolutionsNote2009.pdf. Accessed May 18, 2018.

[15]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, 2015.

[16]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

[18]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett, 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14.

[19]

N. J. Hicks, Notes on Differential Geometry, van Nostrand, 1965.

[20]

A. IannizzottoS. Mosconi and M. Squassina, Global hölder regularity for the fractional p-Laplacian, Revista Matemática Iberoamericana, 32 (2016), 1353-1392. doi: 10.4171/RMI/921.

[21]

M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Physical Review Letters, 56 (1986), 889. doi: 10.1103/PhysRevLett.56.889.

[22]

J.-M. Lasry and P.-L. Lions, A remark on regularization in Hilbert spaces, Israel Journal of Mathematics, 55 (1986), 257-266. doi: 10.1007/BF02765025.

[23]

A. Porretta and P. Souplet, The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation, International Mathematics Research Notices, 17 (2017), 5260-5301. doi: 10.1093/imrn/rnw154.

[24]

A. Porretta and P. Souplet, Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier, 34 (2017), 1913-1923. doi: 10.1016/j.anihpc.2017.02.001.

[25]

A. Quaas and A. Rodríguez, Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms, Journal of Differential Equations, 264 (2018), 2897-2935. doi: 10.1016/j.jde.2017.11.008.

[26]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Springer Science & Business Media, 2007.

[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal, 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.

[30]

L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace, Indiana University Mathematics Journal, 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.

[31]

P. Souplet, Gradient blow-up for multidimensional nonlinear para-bolic equations with general boundary conditions, Differential and Integral Equations, 15 (2002), 237-256.

[32]

P. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, Journal d'analyse mathématique, 99 (2006), 355-396. doi: 10.1007/BF02789452.

[33]

T. Tabet Tchamba, Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian, Asymptotic Analysis, 66 (2010), 161-186.

[34]

L. Yuxiang and P. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Communications in Mathematical Physics, 293 (2010), 499-517. doi: 10.1007/s00220-009-0936-8.

show all references

References:
[1]

A. Attouchi, Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion, Journal of Differential Equations, 253 (2012), 2474-2492. doi: 10.1016/j.jde.2012.07.002.

[2]

A. Attouchi and P. Souplet, Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with p-Laplacian diffusion, Transactions of the American Mathematical Society, 369 (2017), 935-974. doi: 10.1090/tran/6684.

[3]

G. BarlesE. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana University Mathematics Journal, 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315.

[4]

G. Barles and F. Da Lio, On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations, Journal de Mathématiques Pures et Appliquées, 83 (2004), 53-75. doi: 10.1016/S0021-7824(03)00070-9.

[5]

G. BarlesS. KoikeO. Ley and E. Topp, Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians, Calculus of Variations and Partial Differential Equations, 54 (2015), 539-572. doi: 10.1007/s00526-014-0794-x.

[6]

G. BarlesO. Ley and E. Topp, Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior, Nonlinearity, 30 (2017), 703-734. doi: 10.1088/1361-6544/aa527f.

[7]

G. Barles and E. Topp, Existence, uniqueness, and asymptotic behavior for nonlocal parabolic problems with dominating gradient terms, SIAM Journal on Mathematical Analysis, 48 (2016), 1512-1547. doi: 10.1137/140967192.

[8]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.

[9]

M. Crandall, M. Kocan, P. Soravia and A. Swiech, On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, in Progress in Elliptic and Parabolic Partial Differential Equations, vol. 350, Pitman Research Notes in Mathematics Series, 1996,136-162.

[10]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[11]

G. DávilaA. Quaas and E. Topp, Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms, Mathematische Annalen, 369 (2017), 1211-1236. doi: 10.1007/s00208-016-1481-3.

[12]

L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, Journal of Differential Equations, 263 (2017), 765-778. doi: 10.1016/j.jde.2017.02.051.

[13]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[14]

F. Dragoni, Introduction to Viscosity Solutions for Nonlinear PDEs, Available at https://orca-mwe.cf.ac.uk/75040/1/ViscSolutionsNote2009.pdf. Accessed May 18, 2018.

[15]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, 2015.

[16]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.

[18]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett, 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14.

[19]

N. J. Hicks, Notes on Differential Geometry, van Nostrand, 1965.

[20]

A. IannizzottoS. Mosconi and M. Squassina, Global hölder regularity for the fractional p-Laplacian, Revista Matemática Iberoamericana, 32 (2016), 1353-1392. doi: 10.4171/RMI/921.

[21]

M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces, Physical Review Letters, 56 (1986), 889. doi: 10.1103/PhysRevLett.56.889.

[22]

J.-M. Lasry and P.-L. Lions, A remark on regularization in Hilbert spaces, Israel Journal of Mathematics, 55 (1986), 257-266. doi: 10.1007/BF02765025.

[23]

A. Porretta and P. Souplet, The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation, International Mathematics Research Notices, 17 (2017), 5260-5301. doi: 10.1093/imrn/rnw154.

[24]

A. Porretta and P. Souplet, Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier, 34 (2017), 1913-1923. doi: 10.1016/j.anihpc.2017.02.001.

[25]

A. Quaas and A. Rodríguez, Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms, Journal of Differential Equations, 264 (2018), 2897-2935. doi: 10.1016/j.jde.2017.11.008.

[26]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Springer Science & Business Media, 2007.

[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal, 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.

[30]

L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace, Indiana University Mathematics Journal, 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.

[31]

P. Souplet, Gradient blow-up for multidimensional nonlinear para-bolic equations with general boundary conditions, Differential and Integral Equations, 15 (2002), 237-256.

[32]

P. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, Journal d'analyse mathématique, 99 (2006), 355-396. doi: 10.1007/BF02789452.

[33]

T. Tabet Tchamba, Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian, Asymptotic Analysis, 66 (2010), 161-186.

[34]

L. Yuxiang and P. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Communications in Mathematical Physics, 293 (2010), 499-517. doi: 10.1007/s00220-009-0936-8.

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