March 2018, 38(3): 1405-1425. doi: 10.3934/dcds.2018057

Improved energy methods for nonlocal diffusion problems

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain

Received  May 2017 Revised  September 2017 Published  December 2017

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:
\begin{equation*} Lu (x) := \int_{\mathbb{R}^N} K(x, y) (u(y) -u(x)) \,\mathrm{d} y, \end{equation*}
where
$L$
acts on a real function
$u$
defined on
$\mathbb{R}^N$
, and we assume that
$K(x, y)$
is uniformly strictly positive in a neighbourhood of
$x=y$
. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation
$\partial_t u = L u$
as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the
$L^p$
norms of
$u$
and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.
Citation: José A. Cañizo, Alexis Molino. Improved energy methods for nonlocal diffusion problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1405-1425. doi: 10.3934/dcds.2018057
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems American Mathematical Society; Real Sociedad Matemática Española, 2010, http://www.worldcat.org/isbn/9780821852309.

[2]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of Many-Particle systems: An essay on recent research, Monatshefte für Mathematik, 142 (2004), 35–43, URL http://dx.doi.org/10.1007/s00605-004-0239-2. doi: 10.1007/s00605-004-0239-2.

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX 1983/84 (eds. J. Azéma and M. Yor), vol. 1123 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1985, chapter 13, 177-206, URL http://dx.doi.org/10.1007/bfb0075847.

[4]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.

[5]

C. Brändle and A. de Pablo, Nonlocal heat equations: decay estimates and Nash inequalities, 2015, http://arxiv.org/abs/1312.4661.

[6]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Annales de l'Institute Henri Poincaré. Probabilités et statistiques, 23 (1987), 245–287, URL http://eudml.org/doc/77309.

[7]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1–82, URL http://dx.doi.org/10.1007/s006050170032. doi: 10.1007/s006050170032.

[8]

D. Chafaï, Entropies, convexity, and functional inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325–363, URL http://projecteuclid.org/euclid.kjm/1250283556. doi: 10.1215/kjm/1250283556.

[9]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271–291, URL http://dx.doi.org/10.1016/j.matpur.2006.04.005. doi: 10.1016/j.matpur.2006.04.005.

[10]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, Journal of Differential Equations, 241 (2007), 332–358, URL http://www.sciencedirect.com/science/article/pii/S0022039607002082. doi: 10.1016/j.jde.2007.06.002.

[11]

C. CortázarM. ElguetaJ. García-Melián and S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem, Indiana Univ. Math. J., 60 (2011), 209-232. doi: 10.1512/iumj.2011.60.4385.

[12]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Finite mass solutions for a nonlocal inhomogeneous dispersal equation, Discrete and Continuous Dynamical Systems, 35 (2015), 1409–1419, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10560.

[13]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, An inhomogeneous nonlocal diffusion problem with unbounded steps, Journal of Evolution Equations, 16 (2016), 209–232, URL http://dx.doi.org/10.1007/s00028-015-0299-x. doi: 10.1007/s00028-015-0299-x.

[14]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53–60, URL http://dx.doi.org/10.1007/s11856-009-0019-8. doi: 10.1007/s11856-009-0019-8.

[15]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137–156, URL http://dx.doi.org/10.1007/s00205-007-0062-8.

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2004), 245–316, URL http://dx.doi.org/10.1007/s00222-004-0389-9. doi: 10.1007/s00222-004-0389-9.

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley Series in Probability and Statistics, Wiley, 1986, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471081868.

[18]

M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations vol. 79 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 2010, URL http://dx.doi.org/10.1007/978-0-8176-4651-6, Asymptotic behavior of solutions and self-similar solutions.

[19]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061–1083, URL http://www.jstor.org/stable/2373688. doi: 10.2307/2373688.

[20]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, Journal of Functional Analysis, 251 (2007), 399–437, URL http://dx.doi.org/10.1016/j.jfa.2007.07.013. doi: 10.1016/j.jfa.2007.07.013.

[21]

L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, Journal of Evolution Equations, 8 (2008), 617–629, URL http://dx.doi.org/10.1007/s00028-008-0372-9. doi: 10.1007/s00028-008-0372-9.

[22]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163–187, URL http://dx.doi.org/10.1016/j.matpur.2009.04.009. doi: 10.1016/j.matpur.2009.04.009.

[23]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697–702, URL http://dx.doi.org/10.1016/j.crma.2004.03.006. doi: 10.1016/j.crma.2004.03.006.

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1235–1260, URL http://www.sciencedirect.com/science/article/pii/S0021782405000528. doi: 10.1016/j.matpur.2005.04.001.

[25]

S. Mischler and I. Tristani, Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations, J. Éc. Polytech. Math., 4 (2017), 389–433, URL http://arxiv.org/abs/1507.04861. doi: 10.5802/jep.46.

[26]

A. Molino and J. D. Rossi, Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence, Zeitschrift für angewandte Mathematik und Physik 67 (2016), Art. 41, 14 pp, URL http://dx.doi.org/10.1007/s00033-016-0649-8.

[27]

J. Nash, Continuity of solutions of parabolic and elliptic equations, URL http://dx.doi.org/10.2307/2372841.

[28]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361–400, URL http://dx.doi.org/10.1006/jfan.1999.3557. doi: 10.1006/jfan.1999.3557.

[29]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau-type approximations to the heat equation, SIAM Journal on Applied Mathematics, 73 (2013), 1416–1438, URL http://dx.doi.org/10.1137/120876290. doi: 10.1137/120876290.

[30]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Communications in Partial Differential Equations, 5 (1980), 449–473, URL http://dx.doi.org/10.1080/0360530800882145. doi: 10.1080/0360530800882145.

[31]

J.-W. SunW.-T. Li and F.-Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3501-3509. doi: 10.1016/j.na.2011.02.034.

[32]

G. Toscani, A Rosenau-type approach to the approximation of the linear Fokker-Planck equation, 2017, http://arxiv.org/abs/1703.10909.

[33]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, Amsterdam, Netherlands; Boston, U.S.A., 1 (2002), 71–305, URL http://www.umpa.ens-lyon.fr/~{}cvillani/GZPDF/B01.Handbook.pdf.gz.

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems American Mathematical Society; Real Sociedad Matemática Española, 2010, http://www.worldcat.org/isbn/9780821852309.

[2]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of Many-Particle systems: An essay on recent research, Monatshefte für Mathematik, 142 (2004), 35–43, URL http://dx.doi.org/10.1007/s00605-004-0239-2. doi: 10.1007/s00605-004-0239-2.

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX 1983/84 (eds. J. Azéma and M. Yor), vol. 1123 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1985, chapter 13, 177-206, URL http://dx.doi.org/10.1007/bfb0075847.

[4]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.

[5]

C. Brändle and A. de Pablo, Nonlocal heat equations: decay estimates and Nash inequalities, 2015, http://arxiv.org/abs/1312.4661.

[6]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Annales de l'Institute Henri Poincaré. Probabilités et statistiques, 23 (1987), 245–287, URL http://eudml.org/doc/77309.

[7]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1–82, URL http://dx.doi.org/10.1007/s006050170032. doi: 10.1007/s006050170032.

[8]

D. Chafaï, Entropies, convexity, and functional inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325–363, URL http://projecteuclid.org/euclid.kjm/1250283556. doi: 10.1215/kjm/1250283556.

[9]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271–291, URL http://dx.doi.org/10.1016/j.matpur.2006.04.005. doi: 10.1016/j.matpur.2006.04.005.

[10]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, Journal of Differential Equations, 241 (2007), 332–358, URL http://www.sciencedirect.com/science/article/pii/S0022039607002082. doi: 10.1016/j.jde.2007.06.002.

[11]

C. CortázarM. ElguetaJ. García-Melián and S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem, Indiana Univ. Math. J., 60 (2011), 209-232. doi: 10.1512/iumj.2011.60.4385.

[12]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Finite mass solutions for a nonlocal inhomogeneous dispersal equation, Discrete and Continuous Dynamical Systems, 35 (2015), 1409–1419, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10560.

[13]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, An inhomogeneous nonlocal diffusion problem with unbounded steps, Journal of Evolution Equations, 16 (2016), 209–232, URL http://dx.doi.org/10.1007/s00028-015-0299-x. doi: 10.1007/s00028-015-0299-x.

[14]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53–60, URL http://dx.doi.org/10.1007/s11856-009-0019-8. doi: 10.1007/s11856-009-0019-8.

[15]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137–156, URL http://dx.doi.org/10.1007/s00205-007-0062-8.

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2004), 245–316, URL http://dx.doi.org/10.1007/s00222-004-0389-9. doi: 10.1007/s00222-004-0389-9.

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley Series in Probability and Statistics, Wiley, 1986, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471081868.

[18]

M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations vol. 79 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 2010, URL http://dx.doi.org/10.1007/978-0-8176-4651-6, Asymptotic behavior of solutions and self-similar solutions.

[19]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061–1083, URL http://www.jstor.org/stable/2373688. doi: 10.2307/2373688.

[20]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, Journal of Functional Analysis, 251 (2007), 399–437, URL http://dx.doi.org/10.1016/j.jfa.2007.07.013. doi: 10.1016/j.jfa.2007.07.013.

[21]

L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, Journal of Evolution Equations, 8 (2008), 617–629, URL http://dx.doi.org/10.1007/s00028-008-0372-9. doi: 10.1007/s00028-008-0372-9.

[22]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163–187, URL http://dx.doi.org/10.1016/j.matpur.2009.04.009. doi: 10.1016/j.matpur.2009.04.009.

[23]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697–702, URL http://dx.doi.org/10.1016/j.crma.2004.03.006. doi: 10.1016/j.crma.2004.03.006.

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1235–1260, URL http://www.sciencedirect.com/science/article/pii/S0021782405000528. doi: 10.1016/j.matpur.2005.04.001.

[25]

S. Mischler and I. Tristani, Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations, J. Éc. Polytech. Math., 4 (2017), 389–433, URL http://arxiv.org/abs/1507.04861. doi: 10.5802/jep.46.

[26]

A. Molino and J. D. Rossi, Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence, Zeitschrift für angewandte Mathematik und Physik 67 (2016), Art. 41, 14 pp, URL http://dx.doi.org/10.1007/s00033-016-0649-8.

[27]

J. Nash, Continuity of solutions of parabolic and elliptic equations, URL http://dx.doi.org/10.2307/2372841.

[28]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361–400, URL http://dx.doi.org/10.1006/jfan.1999.3557. doi: 10.1006/jfan.1999.3557.

[29]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau-type approximations to the heat equation, SIAM Journal on Applied Mathematics, 73 (2013), 1416–1438, URL http://dx.doi.org/10.1137/120876290. doi: 10.1137/120876290.

[30]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Communications in Partial Differential Equations, 5 (1980), 449–473, URL http://dx.doi.org/10.1080/0360530800882145. doi: 10.1080/0360530800882145.

[31]

J.-W. SunW.-T. Li and F.-Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3501-3509. doi: 10.1016/j.na.2011.02.034.

[32]

G. Toscani, A Rosenau-type approach to the approximation of the linear Fokker-Planck equation, 2017, http://arxiv.org/abs/1703.10909.

[33]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, Amsterdam, Netherlands; Boston, U.S.A., 1 (2002), 71–305, URL http://www.umpa.ens-lyon.fr/~{}cvillani/GZPDF/B01.Handbook.pdf.gz.

[1]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105

[2]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003

[3]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[4]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[5]

Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315

[6]

Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233

[7]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[8]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099

[9]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241

[10]

Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619

[11]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[12]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206

[13]

María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157

[14]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[15]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

[16]

Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967

[17]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531

[18]

Sergio Grillo, Leandro Salomone, Marcela Zuccalli. On the relationship between the energy shaping and the Lyapunov constraint based methods. Journal of Geometric Mechanics, 2017, 9 (4) : 459-486. doi: 10.3934/jgm.2017018

[19]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799

[20]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (59)
  • HTML views (155)
  • Cited by (0)

Other articles
by authors

[Back to Top]