July  2017, 22(5): 1887-1897. doi: 10.3934/dcdsb.2017112

Long-time behavior of state functions for climate energy balance model

1. 

Institute for Applied System Analysis, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine

2. 

National University of Food Technologies, Kyiv, Ukraine

* Corresponding author: Nataliia V. Gorban

Received  July 2016 Revised  September 2016 Published  March 2017

Fund Project: The authors are partially supported by the Ukrainian State Fund for Fundamental Researches under grant GP/F66/14921, and by the National Academy of Sciences of Ukraine under grant 2284.

We study the long time behavior of state functions for a climate energy balance model (so-called Budyko Model). Existence and properties of weak solutions, and existence of Lyapunov function are obtained. Existence, structure and regularity properties for global and trajectory attractors are justified.

Citation: Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112
References:
[1]

J. M. ArrietaA. Rodrígues-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2695-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

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T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations, Springer, Berlin, 1980. Google Scholar

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F. BalibreaT. CaraballoP. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

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J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations. Nonlinear Science, 7 (1997), 475-502 Erratum, ibid 8: 233,1998. Corrected version appears in Mechanics: from Theory to Computation. Springer Verlag, (2000), 447-474. Google Scholar

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J. M. Ball, Global attractors for damped semilinear wave equations, DCDS, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

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V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976. Google Scholar

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M. I. Budyko, The effects of solar radiation variations, on the climate of the Earth, Tellus, 21 (1969), 611-619.  doi: 10.3402/tellusa.v21i5.10109.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors for 3D Navier-Stokes system, Mathematical Notes, 71 (2002), 177-193.  doi: 10.1023/A:1014190629738.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time, Discrete and Continuous Dynamical Systems, 27 (2010), 1498-1509.  doi: 10.3934/dcds.2010.27.1493.  Google Scholar

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V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete and Continuous Dynamical Systems, 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

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F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc. , New York, 1983. Google Scholar

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H. Díaz and J. Díaz, On a stochastic parabolic PDE arising in climatology, Rev. R. Acad. Cien. Serie A Mat., 96 (2002), 123-128.   Google Scholar

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J. DíazJ. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691.  Google Scholar

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J. DíazJ. Hernández and L. Tello, Some results about multiplicity and bifurcation of stationary solutions of a reaction diffusion climatological model, Rev. R. Acad. Cien. Serie A. Mat., 96 (2002), 357-366.   Google Scholar

[15]

J. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Meth. Appl. Sci., 25 (2002), 327-334.  doi: 10.1002/mma.289.  Google Scholar

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M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, Comptes Rendus de l'Academie des Sciences-Series I -Mathematics, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

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M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications, Applied Mathematics Letters, 39 (2015), 19-21.   Google Scholar

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M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov functions for differential inclusions and applications in physics, biology, and climatology, Continuous and distributed systems Ⅱ. Theory and applications, Series studies in systems. Decis. Control, 30 (2015), 233-243.  doi: 10.1007/978-3-319-19075-4_14.  Google Scholar

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M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov functions for weak solutions of reaction-diffusion equations with discontinuous interaction functions and its applications, Nonautonomous Dyn. Syst., 2 (2015), 1-11.  doi: 10.1515/msds-2015-0001.  Google Scholar

[22]

G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions, Discrete & Continuous Dynamical Systems -Series S, 6 (2013), 387-400.  doi: 10.3934/dcdss.2013.6.387.  Google Scholar

[23]

N. V. GorbanO. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, Theory, Methods and Applications, 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar

[24]

N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2013), 205-220.  doi: 10.1007/978-3-319-03146-0_15.  Google Scholar

[25]

N. V. GorbanO. V. KapustyanP. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2014), 221-237.  doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[26]

N. V. GorbanM. O. GluzmanP. O. Kasyanov and A. M. Tkachuk, Long-Time Behavior of State Functions for Budyko Models, Continuous and distributed systems Ⅲ Series studies in systems. Decis. Control, 69 (2016), 351-359.  doi: 10.1007/978-3-319-40673-2_18.  Google Scholar

[27]

P. Kalita and G. Lukaszewicz, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Analysis: Theory, Methods & Applications, 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[28]

P. Kalita and G. Lukaszewicz, Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions, Nonlinear Analysis: Real World Applications, 19 (2014), 75-88.  doi: 10.1016/j.nonrwa.2014.03.002.  Google Scholar

[29]

O. V. KapustyanP. O. KasyanovJ. Valero and M. Z. Zgurovsky, Sructure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, 211 (2014), 163-180.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar

[30]

O. V. KapustyanP. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Communications on Pure and Applied Analysis, 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[31]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, DCDS-B, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[32]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equation, Ukrainian Mathematical Journal, 55 (2003), 535-547.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

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P. O. KasyanovL. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

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P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar

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P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar

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P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Long-time behaviour of solutions for autonomous evolution hemivariational inequality with multidimensional reaction-displacement law Abstract and Applied Analysis (2012), Art. ID 450984, 21 pp. Google Scholar

[37]

V.S. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[38]

S. Migórski, On the existence of solutions for parabolic hemivariational inequalities, Journal of Computational and Applied Mathematics, 129 (2001), 77-87.  doi: 10.1016/S0377-0427(00)00543-4.  Google Scholar

[39]

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, Journal of Global Optimization, 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar

[40]

F. Morillas and J. Valero, Attractors for reaction-diffusion equation in $R^n$ with continuous nonlinearity, Asymptotic Analysis, 44 (2005), 111-130.   Google Scholar

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M. Otani and H. Fujita, On existence of strong solutions for $\frac{{du}}{{dt}}(t)+\partial\varphi^1(u(t))-\partial\varphi^2(u(t))\ni f(t)$, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, 24 (1977), 575-605. Google Scholar

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P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions Birkhauser, Basel, 1985. Google Scholar

[43]

G. R. Sell and Yu. You, Dynamics of Evolutionary Equations Springer, New York, 2002. Google Scholar

[44]

W. D. Sellers, A global climatic model based on the energy balance of the Earth-atmosphere system, J. Appl. Meteorol., 8 (1989), 392-400.  doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar

[45]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988. Google Scholar

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D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, Siam J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

[47]

D. Terman, A free boundary arising from a model for nerve conduction, J. Diff. Eqs., 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X.  Google Scholar

[48]

J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.  Google Scholar

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J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J Math Anal Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

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M. I. VishikS. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor for a dissipative reaction-diffusion system, Doklady Mathematics, 82 (2010), 869-873.  doi: 10.1134/S1064562410060086.  Google Scholar

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N. V. Zadoianchuk and P. O. Kasyanov, Dynamics of solutions of a class of second-order autonomous evolution inclusions, Cybernetics and Systems Analysis, 48 (2012), 414-428.  doi: 10.1007/s10559-012-9421-z.  Google Scholar

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M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III Springer, Berlin, 2012. Google Scholar

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M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, Continuous and Distributed Systems. Theory and Applications. Solid Mechanics and its Applications, 211 (2014), 149-162.  doi: 10.1007/978-3-319-03146-0_11.  Google Scholar

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M. Z. ZgurovskyP. O. Kasyanov and N. V. Zadoianchuk, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Applied Mathematics Letters, 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.  Google Scholar

show all references

References:
[1]

J. M. ArrietaA. Rodrígues-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2695-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations, Springer, Berlin, 1980. Google Scholar

[3]

F. BalibreaT. CaraballoP. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[4]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations. Nonlinear Science, 7 (1997), 475-502 Erratum, ibid 8: 233,1998. Corrected version appears in Mechanics: from Theory to Computation. Springer Verlag, (2000), 447-474. Google Scholar

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, DCDS, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[6]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976. Google Scholar

[7]

M. I. Budyko, The effects of solar radiation variations, on the climate of the Earth, Tellus, 21 (1969), 611-619.  doi: 10.3402/tellusa.v21i5.10109.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors for 3D Navier-Stokes system, Mathematical Notes, 71 (2002), 177-193.  doi: 10.1023/A:1014190629738.  Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time, Discrete and Continuous Dynamical Systems, 27 (2010), 1498-1509.  doi: 10.3934/dcds.2010.27.1493.  Google Scholar

[10]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete and Continuous Dynamical Systems, 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc. , New York, 1983. Google Scholar

[12]

H. Díaz and J. Díaz, On a stochastic parabolic PDE arising in climatology, Rev. R. Acad. Cien. Serie A Mat., 96 (2002), 123-128.   Google Scholar

[13]

J. DíazJ. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691.  Google Scholar

[14]

J. DíazJ. Hernández and L. Tello, Some results about multiplicity and bifurcation of stationary solutions of a reaction diffusion climatological model, Rev. R. Acad. Cien. Serie A. Mat., 96 (2002), 357-366.   Google Scholar

[15]

J. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Meth. Appl. Sci., 25 (2002), 327-334.  doi: 10.1002/mma.289.  Google Scholar

[16]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, Comptes Rendus de l'Academie des Sciences-Series I -Mathematics, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[17]

E. Feireisl and J. Norbury, Some existence and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh A., 119 (1991), 1-17.  doi: 10.1017/S0308210500028262.  Google Scholar

[18]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1975. Google Scholar

[19]

M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications, Applied Mathematics Letters, 39 (2015), 19-21.   Google Scholar

[20]

M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov functions for differential inclusions and applications in physics, biology, and climatology, Continuous and distributed systems Ⅱ. Theory and applications, Series studies in systems. Decis. Control, 30 (2015), 233-243.  doi: 10.1007/978-3-319-19075-4_14.  Google Scholar

[21]

M. O. GluzmanN. V. Gorban and P. O. Kasyanov, Lyapunov functions for weak solutions of reaction-diffusion equations with discontinuous interaction functions and its applications, Nonautonomous Dyn. Syst., 2 (2015), 1-11.  doi: 10.1515/msds-2015-0001.  Google Scholar

[22]

G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions, Discrete & Continuous Dynamical Systems -Series S, 6 (2013), 387-400.  doi: 10.3934/dcdss.2013.6.387.  Google Scholar

[23]

N. V. GorbanO. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, Theory, Methods and Applications, 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar

[24]

N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2013), 205-220.  doi: 10.1007/978-3-319-03146-0_15.  Google Scholar

[25]

N. V. GorbanO. V. KapustyanP. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2014), 221-237.  doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[26]

N. V. GorbanM. O. GluzmanP. O. Kasyanov and A. M. Tkachuk, Long-Time Behavior of State Functions for Budyko Models, Continuous and distributed systems Ⅲ Series studies in systems. Decis. Control, 69 (2016), 351-359.  doi: 10.1007/978-3-319-40673-2_18.  Google Scholar

[27]

P. Kalita and G. Lukaszewicz, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Analysis: Theory, Methods & Applications, 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[28]

P. Kalita and G. Lukaszewicz, Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions, Nonlinear Analysis: Real World Applications, 19 (2014), 75-88.  doi: 10.1016/j.nonrwa.2014.03.002.  Google Scholar

[29]

O. V. KapustyanP. O. KasyanovJ. Valero and M. Z. Zgurovsky, Sructure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, 211 (2014), 163-180.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar

[30]

O. V. KapustyanP. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Communications on Pure and Applied Analysis, 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[31]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, DCDS-B, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[32]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equation, Ukrainian Mathematical Journal, 55 (2003), 535-547.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

[33]

P. O. KasyanovL. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

[34]

P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar

[35]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar

[36]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Long-time behaviour of solutions for autonomous evolution hemivariational inequality with multidimensional reaction-displacement law Abstract and Applied Analysis (2012), Art. ID 450984, 21 pp. Google Scholar

[37]

V.S. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[38]

S. Migórski, On the existence of solutions for parabolic hemivariational inequalities, Journal of Computational and Applied Mathematics, 129 (2001), 77-87.  doi: 10.1016/S0377-0427(00)00543-4.  Google Scholar

[39]

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, Journal of Global Optimization, 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar

[40]

F. Morillas and J. Valero, Attractors for reaction-diffusion equation in $R^n$ with continuous nonlinearity, Asymptotic Analysis, 44 (2005), 111-130.   Google Scholar

[41]

M. Otani and H. Fujita, On existence of strong solutions for $\frac{{du}}{{dt}}(t)+\partial\varphi^1(u(t))-\partial\varphi^2(u(t))\ni f(t)$, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, 24 (1977), 575-605. Google Scholar

[42]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions Birkhauser, Basel, 1985. Google Scholar

[43]

G. R. Sell and Yu. You, Dynamics of Evolutionary Equations Springer, New York, 2002. Google Scholar

[44]

W. D. Sellers, A global climatic model based on the energy balance of the Earth-atmosphere system, J. Appl. Meteorol., 8 (1989), 392-400.  doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar

[45]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988. Google Scholar

[46]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, Siam J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

[47]

D. Terman, A free boundary arising from a model for nerve conduction, J. Diff. Eqs., 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X.  Google Scholar

[48]

J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.  Google Scholar

[49]

J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J Math Anal Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[50]

M. I. VishikS. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor for a dissipative reaction-diffusion system, Doklady Mathematics, 82 (2010), 869-873.  doi: 10.1134/S1064562410060086.  Google Scholar

[51]

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