American Institute of Mathematical Sciences

Advanced
December  2017, 10(6): 1519-1537. doi: 10.3934/dcdss.2017078

Smooth and non-smooth regularizations of the nonlinear diffusion equation

Giuseppe Tomassetti

Roma Tre University, Engineering Department -Civil Engineering Section, Via Vito Volterra 62, Rome -00152, Italy

Dedicated to Tomáš Roubíček on the occasion of his 60th birthday

Received  March 2016 Revised  August 2016 Published  June 2017

Fund Project: The author is supported by GNFM-INdAM "Gruppo Nazionale per la Fisica Matematica".

We illustrate an alternative derivation of the viscous regularization of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc., 324:331-351]. We propose and discuss a new ''non-smooth'' variant of the viscous regularization and we offer an heuristic argument that indicates that this variant should display interesting hysteretic effects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approximation scheme.

Keywords: Diffusion, backward-parabolic partial differential equations, viscosity, hysteresis.
Mathematics Subject Classification: Primary: 35K65, 74N25; Secondary: 74N30.
Citation: Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078
References:
[1]

P. Atkins and J. de Paula, Atkins' Physical Chemistry W. H. Freeman and Company, New York, 2006. Google Scholar

[2]

G. Barenblatt, M. Bertsch, R. D. Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.  doi: 10.1137/0524082.  Google Scholar

[3]

M. Bertsch, P. Podio-Guidugli and V. Valente, On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pur. Appl., 179 (2001), 331-360.  doi: 10.1007/BF02505962.  Google Scholar

[4]

M. Bertsch, F. Smarrazzo and A. Tesei, Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Analysis PDE, 6 (2013), 1719-1754.  doi: 10.2140/apde.2013.6.1719.  Google Scholar

[5]

E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661, arXiv: 1508.03225. doi: 10.1142/S0218202517500129.  Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[7]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963), 167-178.  doi: 10.1007/BF01262690.  Google Scholar

[8]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.  doi: 10.1137/110828526.  Google Scholar

[9]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Diff. Eq., 254 (2013), 4217-4244.  doi: 10.1016/j.jde.2013.02.014.  Google Scholar

[10]

F. P. Duda and G. Tomassetti, On the effect of elastic distortions on the kinetics of diffusion-induced phase transformations, J. Elasticity, 122 (2016), 179-195.  doi: 10.1007/s10659-015-9539-0.  Google Scholar

[11]

C. Elliot and S. Luckhaus, Generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy, Preprint 887, Institute for Mathematics and its Applications, Minneapolis, 1991. Google Scholar

[12]

C. Elliott and A. Stuart, Viscous Cahn-Hilliard Equation Ⅱ. Analysis, J. Diff. Eq., 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101.  Google Scholar

[13]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[14]

L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.  doi: 10.1142/S0218202504003763.  Google Scholar

[15]

P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10 (1942), 51. Google Scholar

[16]

E. Fried and M. Gurtin, Coherent solid-state phase transitions with atomic diffusion: A thermomechanical treatment, J. Stat. Phys., 95 (1999), 1361-1427.  doi: 10.1023/A:1004535408168.  Google Scholar

[17]

E. Fried and S. Sellers, Microforces and the theory of solute transport, Z. angew. Math. Phys, 51 (2000), 732-751.  doi: 10.1007/PL00001517.  Google Scholar

[18]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[19]

M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua Cambridge University Press, New York, 2010. Google Scholar

[20]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires Dunod, 1969.  Google Scholar

[21]

A. Lucantonio, P. Nardinocchi and L. Teresi, Transient analysis of swelling-induced large deformations in polymer gels, J. Mech. Phys. Solids, 61 (2013), 205-218.  doi: 10.1016/j.jmps.2012.07.010.  Google Scholar

[22]

A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, Compt. Rend. Acad. Sci. -Ser. I-Math., 328 (1999), 1247-1252.  doi: 10.1016/S0764-4442(99)80448-0.  Google Scholar

[23]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asympt. Anal., 22 (2000), 235-259.   Google Scholar

[24]

A. Miranville, A. Pietrus and J.-M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asympt. Anal., 16 (1998), 315-345.   Google Scholar

[25]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, In Material instabilities in continuum mechanics (Edinburgh, 1985-1986), pages 329-342. Oxford University Press, 1988.  Google Scholar

[26]

A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.  doi: 10.1090/S0002-9947-1991-1015926-7.  Google Scholar

[27]

P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Eq., 30 (1994), 614-622.   Google Scholar

[28]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  doi: 10.1007/s11587-006-0008-8.  Google Scholar

[29]

M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.  doi: 10.1007/s00205-013-0666-0.  Google Scholar

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Springer Basel, 2013. Google Scholar

[31]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst.-B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.  Google Scholar

[32]

T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modelling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572.  doi: 10.1007/s00033-015-0566-2.  Google Scholar

[33]

B. E. Sar, S. Fréour, P. Davies and F. Jacquemin, Accounting for differential swelling in the multi-physics modelling of the diffusive behaviour of polymers, ZAMM Z. Angew. Math. Mech., 94 (2014), 452-460.  doi: 10.1002/zamm.201200272.  Google Scholar

[34]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[35]

B. L. T. Thanh, F. Smarrazzo and A. Tesei, Passage to the limit over small parameters in the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 420 (2014), 1265-1300.  doi: 10.1016/j.jmaa.2014.06.036.  Google Scholar

[36]

B. L. T. Thanh, F. Smarrazzo and A. Tesei, Sobolev regularization of a class of forward-backward parabolic equations, J. Diff. Eq., 257 (2014), 1403-1456.  doi: 10.1016/j.jde.2014.05.004.  Google Scholar

[37]

P. Victor, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Part. Diff. Eq., 23 (1998), 457-486.  doi: 10.1080/03605309808821353.  Google Scholar

[38]

A. Visintin, Differential Models of Hysteresis Springer Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[39]

A. Visintin, Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Diff. Eq., 15 (2002), 115-132.  doi: 10.1007/s005260100120.  Google Scholar

show all references

References:
[1]

P. Atkins and J. de Paula, Atkins' Physical Chemistry W. H. Freeman and Company, New York, 2006. Google Scholar

[2]

G. Barenblatt, M. Bertsch, R. D. Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.  doi: 10.1137/0524082.  Google Scholar

[3]

M. Bertsch, P. Podio-Guidugli and V. Valente, On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pur. Appl., 179 (2001), 331-360.  doi: 10.1007/BF02505962.  Google Scholar

[4]

M. Bertsch, F. Smarrazzo and A. Tesei, Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Analysis PDE, 6 (2013), 1719-1754.  doi: 10.2140/apde.2013.6.1719.  Google Scholar

[5]

E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661, arXiv: 1508.03225. doi: 10.1142/S0218202517500129.  Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[7]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963), 167-178.  doi: 10.1007/BF01262690.  Google Scholar

[8]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.  doi: 10.1137/110828526.  Google Scholar

[9]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Diff. Eq., 254 (2013), 4217-4244.  doi: 10.1016/j.jde.2013.02.014.  Google Scholar

[10]

F. P. Duda and G. Tomassetti, On the effect of elastic distortions on the kinetics of diffusion-induced phase transformations, J. Elasticity, 122 (2016), 179-195.  doi: 10.1007/s10659-015-9539-0.  Google Scholar

[11]

C. Elliot and S. Luckhaus, Generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy, Preprint 887, Institute for Mathematics and its Applications, Minneapolis, 1991. Google Scholar

[12]

C. Elliott and A. Stuart, Viscous Cahn-Hilliard Equation Ⅱ. Analysis, J. Diff. Eq., 128 (1996), 387-414.  doi: 10.1006/jdeq.1996.0101.  Google Scholar

[13]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[14]

L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.  doi: 10.1142/S0218202504003763.  Google Scholar

[15]

P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10 (1942), 51. Google Scholar

[16]

E. Fried and M. Gurtin, Coherent solid-state phase transitions with atomic diffusion: A thermomechanical treatment, J. Stat. Phys., 95 (1999), 1361-1427.  doi: 10.1023/A:1004535408168.  Google Scholar

[17]

E. Fried and S. Sellers, Microforces and the theory of solute transport, Z. angew. Math. Phys, 51 (2000), 732-751.  doi: 10.1007/PL00001517.  Google Scholar

[18]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[19]

M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua Cambridge University Press, New York, 2010. Google Scholar

[20]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires Dunod, 1969.  Google Scholar

[21]

A. Lucantonio, P. Nardinocchi and L. Teresi, Transient analysis of swelling-induced large deformations in polymer gels, J. Mech. Phys. Solids, 61 (2013), 205-218.  doi: 10.1016/j.jmps.2012.07.010.  Google Scholar

[22]

A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance, Compt. Rend. Acad. Sci. -Ser. I-Math., 328 (1999), 1247-1252.  doi: 10.1016/S0764-4442(99)80448-0.  Google Scholar

[23]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asympt. Anal., 22 (2000), 235-259.   Google Scholar

[24]

A. Miranville, A. Pietrus and J.-M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asympt. Anal., 16 (1998), 315-345.   Google Scholar

[25]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, In Material instabilities in continuum mechanics (Edinburgh, 1985-1986), pages 329-342. Oxford University Press, 1988.  Google Scholar

[26]

A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.  doi: 10.1090/S0002-9947-1991-1015926-7.  Google Scholar

[27]

P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Eq., 30 (1994), 614-622.   Google Scholar

[28]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  doi: 10.1007/s11587-006-0008-8.  Google Scholar

[29]

M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.  doi: 10.1007/s00205-013-0666-0.  Google Scholar

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Springer Basel, 2013. Google Scholar

[31]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst.-B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.  Google Scholar

[32]

T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modelling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572.  doi: 10.1007/s00033-015-0566-2.  Google Scholar

[33]

B. E. Sar, S. Fréour, P. Davies and F. Jacquemin, Accounting for differential swelling in the multi-physics modelling of the diffusive behaviour of polymers, ZAMM Z. Angew. Math. Mech., 94 (2014), 452-460.  doi: 10.1002/zamm.201200272.  Google Scholar

[34]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[35]

B. L. T. Thanh, F. Smarrazzo and A. Tesei, Passage to the limit over small parameters in the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 420 (2014), 1265-1300.  doi: 10.1016/j.jmaa.2014.06.036.  Google Scholar

[36]

B. L. T. Thanh, F. Smarrazzo and A. Tesei, Sobolev regularization of a class of forward-backward parabolic equations, J. Diff. Eq., 257 (2014), 1403-1456.  doi: 10.1016/j.jde.2014.05.004.  Google Scholar

[37]

P. Victor, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Part. Diff. Eq., 23 (1998), 457-486.  doi: 10.1080/03605309808821353.  Google Scholar

[38]

A. Visintin, Differential Models of Hysteresis Springer Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[39]

A. Visintin, Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Diff. Eq., 15 (2002), 115-132.  doi: 10.1007/s005260100120.  Google Scholar

Figure 1.  (a) plots of $A z(s)$ and $K v(s)$; (b) parametric plot of $s\mapsto (A z(s),Kv(s))$. Here $s=t/\tau$ is the rescaled time.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Plots of the free energy in (105) and of its derivative (respectively, solid and dashed line).
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[2]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[3]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[4]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[5]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[6]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189
[7]

Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338
[8]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[9]

Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047
[10]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[11]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[12]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[13]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[14]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[15]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[16]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[17]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[18]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[19]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[20]

Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (19)
  • HTML views (65)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences