February  2013, 6(1): 257-275. doi: 10.3934/dcdss.2013.6.257

Structural stability of rate-independent nonpotential flows

1. 

Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050, Povo di Trento, Italy

Received  April 2011 Revised  August 2011 Published  October 2012

Several phenomena may be represented by doubly-nonlinear equations of the form $$ \alpha(D_tu) - \nabla\cdot \gamma(\nabla u)\ni h, $$ with $\alpha$ and $\gamma$ (possibly multivalued) maximal monotone mappings. Hysteresis effects are characterized by rate-independence, which corresponds to $\alpha$ positively homogeneous of zero degree.
    Fitzpatrick showed that any maximal monotone relation may be represented variationally. On this basis, an initial- and boundary-value problem associated to the equation above is here formulated as a null-minimization problem, without assuming $\gamma$ to be cyclically monotone. Existence of a solution $u\in H^1(0,T; H^1(\Omega))$ is proved, as well as its stability with respect to variations of the data, of the mapping $\gamma$, and of the domain $\Omega$.
Citation: Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
References:
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H. Attouch, "Variational Convergence for Functions and Operators,", Pitman, (1984).   Google Scholar

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G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, Differential Integral Equations, 6 (1993), 1161.   Google Scholar

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V. Barbu, Existence theorems for a class of two point boundary problems,, J. Differential Equations, 17 (1975), 236.   Google Scholar

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V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010).   Google Scholar

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V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", Editura Academiei, (1978).   Google Scholar

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A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford University Press, (1998).   Google Scholar

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H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973).   Google Scholar

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H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).   Google Scholar

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H. Brezis and I. Ekeland, Un principe variationnel associé àcertaines équations parabo-liques.I. Le cas indépendant du temps,, II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971.   Google Scholar

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M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

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M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Anal., 15 (2008), 87.   Google Scholar

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R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, Set-Valued Analysis, 10 (2002), 297.  doi: 10.1023/A:1020639314056.  Google Scholar

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R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379.  doi: 10.1090/S0002-9939-03-07053-9.  Google Scholar

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P. Colli and A. Visintin, On a class of doubly nonlinear evolution problems,, Communications in P. D. E. s, 15 (1990), 737.   Google Scholar

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G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).   Google Scholar

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G. Dal Maso and R. Toader, A model for the quasi-static growth or brittle fractures: Existence and approximation results,, Arch. Rat. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

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E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842.   Google Scholar

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W. Fenchel, "Convex Cones, Sets, and Functions,", Princeton Univ., (1953).   Google Scholar

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S. Fitzpatrick, Representing monotone operators by convex functions,, Workshop/Minicon-ference on Functional Analysis and Optimization (Canberra, (1988), 59.   Google Scholar

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G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem: Existence and approximation results,, J. Mech. Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

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H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operator Gleichungen und Operator Differential Gleichungen,", Akademie-Verlag, (1974).   Google Scholar

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H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum,, Math. Nachr., 57 (1973), 127.  doi: 10.1002/mana.19730570107.  Google Scholar

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N. Ghoussoub, A variational theory for monotone vector fields,, J. Fixed Point Theory Appl., 4 (2008), 107.  doi: 10.1007/s11784-008-0083-4.  Google Scholar

[27]

N. Ghoussoub, "Selfdual Partial Differential Systems and Their Variational Principles,", Springer, (2009).   Google Scholar

[28]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, Math. Ann., 330 (2004), 519.  doi: 10.1007/s00208-004-0558-6.  Google Scholar

[29]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Springer, (1999).   Google Scholar

[30]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", North-Holland, (1979).   Google Scholar

[31]

H. Jian, On the homogenization of degenerate parabolic equations,, Acta Math. Appl. Sinica, 16 (2000), 100.  doi: 10.1007/BF02670970.  Google Scholar

[32]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989).   Google Scholar

[33]

P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations,", Gakkotosho, (1997).   Google Scholar

[34]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).   Google Scholar

[35]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vols. I, (1972).   Google Scholar

[36]

J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, Set-Valued Anal., 13 (2005), 21.  doi: 10.1007/s11228-004-4170-4.  Google Scholar

[37]

J. E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, Proc. Amer. Math. Soc., 136 (2008), 873.  doi: 10.1090/S0002-9939-07-09176-9.  Google Scholar

[38]

J. E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, J. Nonlinear Convex Anal., 2 (2001), 243.   Google Scholar

[39]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003).   Google Scholar

[40]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461.   Google Scholar

[41]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl., 11 (2004), 151.   Google Scholar

[42]

A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Arch. Rational Mech. Anal., 162 (2002), 137.  doi: 10.1007/s002050200194.  Google Scholar

[43]

A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation,, Electron. J. Differential Equations, 1 (2001).   Google Scholar

[44]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976).   Google Scholar

[45]

J. P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, C. R. Math. Acad. Sci. Paris, 338 (2004), 853.   Google Scholar

[46]

J. P. Penot, The relevance of convex analysis for the study of monotonicity,, Nonlinear Anal., 58 (2004), 855.  doi: 10.1016/j.na.2004.05.018.  Google Scholar

[47]

R. T. Rockafellar, A general correspondence between dual minimax problems and convex programs,, Pacific J. Math., 25 (1968), 597.   Google Scholar

[48]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969).   Google Scholar

[49]

R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 97.   Google Scholar

[50]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005).   Google Scholar

[51]

G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations,, Discrete Contin. Dyn. Syst., 18 (2007), 15.  doi: 10.3934/dcds.2007.18.15.  Google Scholar

[52]

T. Senba, On some nonlinear evolution equations,, Funkcial. Ekvac., 29 (1986), 243.   Google Scholar

[53]

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

[54]

U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations,, S. I. A. M. J. Control Optim., 8 (2008), 1615.  doi: 10.1137/070684574.  Google Scholar

[55]

A. Visintin, A phase transition problem with delay,, Control and Cybernetics, 11 (1982), 5.   Google Scholar

[56]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).   Google Scholar

[57]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996).   Google Scholar

[58]

A. Visintin, Quasilinear hyperbolic equations with hysteresis,, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451.   Google Scholar

[59]

A. Visintin, Maxwell's equations with vector hysteresis,, Arch. Rat. Mech. Anal., 175 (2005), 1.  doi: 10.1007/s00205-004-0333-6.  Google Scholar

[60]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, Adv. Math. Sci. Appl., 18 (2008), 633.   Google Scholar

[61]

A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization,, Boll. Un. Mat. Ital., III (2010), 591.   Google Scholar

[62]

A. Visintin, Structural stability of doubly-nonlinear flows,, Boll. Un. Mat. Ital., IV (2011), 363.   Google Scholar

[63]

A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().   Google Scholar

show all references

References:
[1]

S. Aizicovici and Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations,, Panamer. Math. J, 7 (1997), 1.   Google Scholar

[2]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitman, (1984).   Google Scholar

[3]

G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, Differential Integral Equations, 6 (1993), 1161.   Google Scholar

[4]

V. Barbu, Existence theorems for a class of two point boundary problems,, J. Differential Equations, 17 (1975), 236.   Google Scholar

[5]

V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010).   Google Scholar

[6]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", Editura Academiei, (1978).   Google Scholar

[7]

A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford University Press, (1998).   Google Scholar

[8]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973).   Google Scholar

[9]

H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).   Google Scholar

[10]

H. Brezis and I. Ekeland, Un principe variationnel associé àcertaines équations parabo-liques.I. Le cas indépendant du temps,, II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971.   Google Scholar

[11]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[12]

M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Anal., 15 (2008), 87.   Google Scholar

[13]

R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, Set-Valued Analysis, 10 (2002), 297.  doi: 10.1023/A:1020639314056.  Google Scholar

[14]

R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379.  doi: 10.1090/S0002-9939-03-07053-9.  Google Scholar

[15]

P. Colli, On some doubly nonlinear evolution equations in Banach spaces,, Japan J. Indust. Appl. Math., 9 (1992), 181.  doi: 10.1007/BF03167565.  Google Scholar

[16]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution problems,, Communications in P. D. E. s, 15 (1990), 737.   Google Scholar

[17]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).   Google Scholar

[18]

G. Dal Maso and R. Toader, A model for the quasi-static growth or brittle fractures: Existence and approximation results,, Arch. Rat. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

[19]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842.   Google Scholar

[20]

I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles,", Dunod Gau-thier-Villars, (1974).   Google Scholar

[21]

W. Fenchel, "Convex Cones, Sets, and Functions,", Princeton Univ., (1953).   Google Scholar

[22]

S. Fitzpatrick, Representing monotone operators by convex functions,, Workshop/Minicon-ference on Functional Analysis and Optimization (Canberra, (1988), 59.   Google Scholar

[23]

G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem: Existence and approximation results,, J. Mech. Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[24]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operator Gleichungen und Operator Differential Gleichungen,", Akademie-Verlag, (1974).   Google Scholar

[25]

H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum,, Math. Nachr., 57 (1973), 127.  doi: 10.1002/mana.19730570107.  Google Scholar

[26]

N. Ghoussoub, A variational theory for monotone vector fields,, J. Fixed Point Theory Appl., 4 (2008), 107.  doi: 10.1007/s11784-008-0083-4.  Google Scholar

[27]

N. Ghoussoub, "Selfdual Partial Differential Systems and Their Variational Principles,", Springer, (2009).   Google Scholar

[28]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, Math. Ann., 330 (2004), 519.  doi: 10.1007/s00208-004-0558-6.  Google Scholar

[29]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Springer, (1999).   Google Scholar

[30]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", North-Holland, (1979).   Google Scholar

[31]

H. Jian, On the homogenization of degenerate parabolic equations,, Acta Math. Appl. Sinica, 16 (2000), 100.  doi: 10.1007/BF02670970.  Google Scholar

[32]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989).   Google Scholar

[33]

P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations,", Gakkotosho, (1997).   Google Scholar

[34]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).   Google Scholar

[35]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vols. I, (1972).   Google Scholar

[36]

J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, Set-Valued Anal., 13 (2005), 21.  doi: 10.1007/s11228-004-4170-4.  Google Scholar

[37]

J. E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, Proc. Amer. Math. Soc., 136 (2008), 873.  doi: 10.1090/S0002-9939-07-09176-9.  Google Scholar

[38]

J. E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, J. Nonlinear Convex Anal., 2 (2001), 243.   Google Scholar

[39]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003).   Google Scholar

[40]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461.   Google Scholar

[41]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl., 11 (2004), 151.   Google Scholar

[42]

A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Arch. Rational Mech. Anal., 162 (2002), 137.  doi: 10.1007/s002050200194.  Google Scholar

[43]

A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation,, Electron. J. Differential Equations, 1 (2001).   Google Scholar

[44]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976).   Google Scholar

[45]

J. P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, C. R. Math. Acad. Sci. Paris, 338 (2004), 853.   Google Scholar

[46]

J. P. Penot, The relevance of convex analysis for the study of monotonicity,, Nonlinear Anal., 58 (2004), 855.  doi: 10.1016/j.na.2004.05.018.  Google Scholar

[47]

R. T. Rockafellar, A general correspondence between dual minimax problems and convex programs,, Pacific J. Math., 25 (1968), 597.   Google Scholar

[48]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969).   Google Scholar

[49]

R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 97.   Google Scholar

[50]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005).   Google Scholar

[51]

G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations,, Discrete Contin. Dyn. Syst., 18 (2007), 15.  doi: 10.3934/dcds.2007.18.15.  Google Scholar

[52]

T. Senba, On some nonlinear evolution equations,, Funkcial. Ekvac., 29 (1986), 243.   Google Scholar

[53]

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

[54]

U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations,, S. I. A. M. J. Control Optim., 8 (2008), 1615.  doi: 10.1137/070684574.  Google Scholar

[55]

A. Visintin, A phase transition problem with delay,, Control and Cybernetics, 11 (1982), 5.   Google Scholar

[56]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).   Google Scholar

[57]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996).   Google Scholar

[58]

A. Visintin, Quasilinear hyperbolic equations with hysteresis,, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451.   Google Scholar

[59]

A. Visintin, Maxwell's equations with vector hysteresis,, Arch. Rat. Mech. Anal., 175 (2005), 1.  doi: 10.1007/s00205-004-0333-6.  Google Scholar

[60]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, Adv. Math. Sci. Appl., 18 (2008), 633.   Google Scholar

[61]

A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization,, Boll. Un. Mat. Ital., III (2010), 591.   Google Scholar

[62]

A. Visintin, Structural stability of doubly-nonlinear flows,, Boll. Un. Mat. Ital., IV (2011), 363.   Google Scholar

[63]

A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().   Google Scholar

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A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

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