American Institute of Mathematical Sciences

July  2017, 22(5): 1909-1964. doi: 10.3934/dcdsb.2017114

Pullback attractors of reaction-diffusion inclusions with space-dependent delay

 1 School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan 430074, China 2 Applied Mathematics, RheinMain University of Applied Sciences, 65197 Wiesbaden, Germany

Received  March 2015 Revised  April 2016 Published  March 2017

Fund Project: Partially supported by DFG grant KL 1203/7-1, the Spanish Ministerio de Ciencia e Innovación project MTM2011-22411, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under the Ayuda 2009/FQM314 and the Proyecto de Excelencia P12-FQM-1492

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

Citation: Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114
References:
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Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.Google Scholar [7] T. A. Burton, Volterra Integral and Differential Equations, volume 202 of Mathematics in Science and Engineering, Elsevier B. V. , Amsterdam, second edition, 2005.Google Scholar [8] T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17. Google Scholar [9] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar [10] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar [11] A. N. Carvalho, J. A. Langa, and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, volume 182 of Applied Mathematical Sciences, Springer, New York, 2013.Google Scholar [12] C. Castaing, L. A. Faik and A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay, Int. J. Appl. Math., 2 (2000), 1005-1026. Google Scholar [13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, volume 580 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1977.Google Scholar [14] V. V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville and V. Pata, Trajectory and global attractors for evolution equations with memory, Appl. Math. Lett., 19 (2006), 87-96. doi: 10.1016/j.aml.2005.03.007. Google Scholar [15] V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142. Google Scholar [16] V. V. Chepyzhov and A. Miranville, On trajectory and global attractors for semilinear heat equations with fading memory, Indiana Univ. Math. J., 55 (2006), 119-167. doi: 10.1512/iumj.2006.55.2597. Google Scholar [17] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl.(9), 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3. Google Scholar [18] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002.Google Scholar [19] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar [20] E. B. Davies, Heat Kernels and Spectral Theory, volume 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989.Google Scholar [21] M. M. Deza and E. Deza, Encyclopedia of Distances, Springer, Heidelberg, third edition, 2014.Google Scholar [22] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. -O. Walther, Delay Equations, volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York, 1995. Functional, complex, and nonlinear analysis.Google Scholar [23] J. Diestel and J. J. Uhl, Jr, Vector Measures, volume 15 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, R. I. , 1977.Google Scholar [24] K. -J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, volume 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.Google Scholar [25] J. R. Giles, Introduction to the Analysis of Metric Spaces, volume 3 of Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge, 1987.Google Scholar [26] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004. Google Scholar [27] N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain, In Continuous and distributed systems, volume 211 of Solid Mech. Appl. , pages 205-220. Springer, Cham, 2013.Google Scholar [28] J. W. Green and F. A. Valentine, On the Arzelá-Ascoli theorem, Math. Mag., 34 (1960/61), 199-202. doi: 10.2307/2687984. Google Scholar [29] G. Haddad, Functional viability theorems for differential inclusions with memory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 179-204. Google Scholar [30] Y. Hino, S. Murakami, and T. Naito, Functional-differential Equations with Infinite Delay, volume 1473 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.Google Scholar [31] N. R. Howes, Modern Analysis and Topology, Universitext. Springer-Verlag, New York, 1995.Google Scholar [32] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. Ⅰ, volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. Theory.Google Scholar [33] A. V. Kapustyan, V. S. Mel'nik, J. Valero, and V. Yasinsky, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, Kyiv, 2008.Google Scholar [34] O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2723-2734. doi: 10.1142/S0218127410027313. Google Scholar [35] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [36] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735. Google Scholar [37] P. E. Kloeden, T. Lorenz and M. Yang, Reaction-diffusion equations with a switched-off reaction zone, Commun. Pure Appl. Anal., 13 (2014), 1907-1933. doi: 10.3934/cpaa.2014.13.1907. Google Scholar [38] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, volume 176 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011.Google Scholar [39] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I. , 1968.Google Scholar [40] V. Lakshmikantham, L. Z. Wen and B. G. Zhang, Theory of Differential Equations with Unbounded Delay, volume 298 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1994.Google Scholar [41] Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029. doi: 10.1016/j.amc.2006.11.187. Google Scholar [42] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. , Inc. , River Edge, NJ, 1996.Google Scholar [43] T. Lorenz, Mutational Analysis, volume 1996 of Lecture Notes in Mathematics. SpringerVerlag, Berlin, 2010. A joint framework for Cauchy problems in and beyond vector spaces.Google Scholar [44] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [45] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. Google Scholar [46] C. B. Morrey, Jr, Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc. , New York, 1966.Google Scholar [47] A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33-44. Google Scholar [48] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.Google Scholar [49] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.Google Scholar [50] B. Schmalfuss, Attractors for the non-autonomous dynamical systems, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 684-689. World Sci. Publ. , River Edge, NJ, 2000.Google Scholar [51] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [52] H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215. doi: 10.1016/j.jmaa.2006.02.041. Google Scholar [53] C.-Y. Sun and C.-K. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034. Google Scholar [54] Y. Wang, C. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems, Discrete Contin. Dyn. Syst., 16 (2006), 587-614. doi: 10.3934/dcds.2006.16.587. Google Scholar [55] Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005. Google Scholar [56] Y. Wang and S. Zhou, Kernel sections on multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Appl. Math., 67 (2009), 343-378. doi: 10.1090/S0033-569X-09-01150-0. Google Scholar [57] K. Yosida, Functional Analysis, volume 123 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin-New York, sixth edition, 1980.Google Scholar [58] M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in nonsmooth systems with applications for earth data processing: Uniform trajectory attractors for nonautonomous evolution inclusions solutions with pointwise pseudomonotone mappings, In Advances in global optimization, volume 95 of Springer Proc. Math. Stat. , pages 283-294. Springer, Cham, 2015.Google Scholar [59] C. Zhao and S. Zhou, Pullback trajectory attractors for evolution equations and application to 3D incompressible non-Newtonian fluid, Nonlinearity, 21 (2008), 1691-1717. doi: 10.1088/0951-7715/21/8/002. Google Scholar [60] C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. Google Scholar [61] W. P. Ziemer, Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989.Google Scholar

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References:
 [1] J. -P. Aubin and A. Cellina, Differential Inclusions, volume 264 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory.Google Scholar [2] J. -P. Aubin and H. Frankowska, Set-valued Analysis, volume 2 of Systems & Control: Foundations & Applications, Birkhäuser Boston Inc. , Boston, MA, 1990.Google Scholar [3] J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. doi: 10.2307/2041821. Google Scholar [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste Romȃnia, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian.Google Scholar [5] P. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972), A47-A50. Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.Google Scholar [7] T. A. Burton, Volterra Integral and Differential Equations, volume 202 of Mathematics in Science and Engineering, Elsevier B. V. , Amsterdam, second edition, 2005.Google Scholar [8] T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17. Google Scholar [9] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar [10] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar [11] A. N. Carvalho, J. A. Langa, and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, volume 182 of Applied Mathematical Sciences, Springer, New York, 2013.Google Scholar [12] C. Castaing, L. A. Faik and A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay, Int. J. Appl. Math., 2 (2000), 1005-1026. Google Scholar [13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, volume 580 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1977.Google Scholar [14] V. V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville and V. Pata, Trajectory and global attractors for evolution equations with memory, Appl. Math. Lett., 19 (2006), 87-96. doi: 10.1016/j.aml.2005.03.007. Google Scholar [15] V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142. Google Scholar [16] V. V. Chepyzhov and A. Miranville, On trajectory and global attractors for semilinear heat equations with fading memory, Indiana Univ. Math. J., 55 (2006), 119-167. doi: 10.1512/iumj.2006.55.2597. Google Scholar [17] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl.(9), 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3. Google Scholar [18] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002.Google Scholar [19] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar [20] E. B. Davies, Heat Kernels and Spectral Theory, volume 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989.Google Scholar [21] M. M. Deza and E. Deza, Encyclopedia of Distances, Springer, Heidelberg, third edition, 2014.Google Scholar [22] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. -O. Walther, Delay Equations, volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York, 1995. Functional, complex, and nonlinear analysis.Google Scholar [23] J. Diestel and J. J. Uhl, Jr, Vector Measures, volume 15 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, R. I. , 1977.Google Scholar [24] K. -J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, volume 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.Google Scholar [25] J. R. Giles, Introduction to the Analysis of Metric Spaces, volume 3 of Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge, 1987.Google Scholar [26] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004. Google Scholar [27] N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain, In Continuous and distributed systems, volume 211 of Solid Mech. Appl. , pages 205-220. Springer, Cham, 2013.Google Scholar [28] J. W. Green and F. A. Valentine, On the Arzelá-Ascoli theorem, Math. Mag., 34 (1960/61), 199-202. doi: 10.2307/2687984. Google Scholar [29] G. Haddad, Functional viability theorems for differential inclusions with memory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 179-204. Google Scholar [30] Y. Hino, S. Murakami, and T. Naito, Functional-differential Equations with Infinite Delay, volume 1473 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.Google Scholar [31] N. R. Howes, Modern Analysis and Topology, Universitext. Springer-Verlag, New York, 1995.Google Scholar [32] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. Ⅰ, volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. Theory.Google Scholar [33] A. V. Kapustyan, V. S. Mel'nik, J. Valero, and V. Yasinsky, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, Kyiv, 2008.Google Scholar [34] O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2723-2734. doi: 10.1142/S0218127410027313. Google Scholar [35] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [36] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735. Google Scholar [37] P. E. Kloeden, T. Lorenz and M. Yang, Reaction-diffusion equations with a switched-off reaction zone, Commun. Pure Appl. Anal., 13 (2014), 1907-1933. doi: 10.3934/cpaa.2014.13.1907. Google Scholar [38] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, volume 176 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011.Google Scholar [39] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I. , 1968.Google Scholar [40] V. Lakshmikantham, L. Z. Wen and B. G. Zhang, Theory of Differential Equations with Unbounded Delay, volume 298 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1994.Google Scholar [41] Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029. doi: 10.1016/j.amc.2006.11.187. Google Scholar [42] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. , Inc. , River Edge, NJ, 1996.Google Scholar [43] T. Lorenz, Mutational Analysis, volume 1996 of Lecture Notes in Mathematics. SpringerVerlag, Berlin, 2010. A joint framework for Cauchy problems in and beyond vector spaces.Google Scholar [44] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [45] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. Google Scholar [46] C. B. Morrey, Jr, Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc. , New York, 1966.Google Scholar [47] A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33-44. Google Scholar [48] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.Google Scholar [49] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.Google Scholar [50] B. Schmalfuss, Attractors for the non-autonomous dynamical systems, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 684-689. World Sci. Publ. , River Edge, NJ, 2000.Google Scholar [51] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [52] H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215. doi: 10.1016/j.jmaa.2006.02.041. Google Scholar [53] C.-Y. Sun and C.-K. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034. Google Scholar [54] Y. Wang, C. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems, Discrete Contin. Dyn. Syst., 16 (2006), 587-614. doi: 10.3934/dcds.2006.16.587. 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