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Regular solutions and global attractors for reaction-diffusion systems without uniqueness
Reaction-diffusion equations with a switched--off reaction zone
1. | Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main |
2. | Institute of Mathematics, Johann Wolfgang Goethe University, 60054 Frankfurt (Main) |
3. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074 |
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Spaces, second edition, Elsevier, Amsterdam, 2003. |
[2] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. |
[3] |
J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory, Springer, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[4] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Noordhoff International Publishing, 1976. |
[5] |
Ph. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972), A47-A50. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[7] |
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. |
[8] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, 1977. |
[9] |
Xinfu Chen, J.-S. Guo and Bei Hu, Dead-core rates for the porous medium equation with a strong absorption, Discrete and Continuous Dynamical Systems, Series B, to appear.
doi: 10.3934/dcdsb.2012.17.1761. |
[10] |
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511566158. |
[11] |
J. Diestel and Jr. J. J. Uhl, Vector Measures, American Mathematical Society, Providence, 1977. |
[12] |
A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math., 57 (2000), 203-217. |
[13] |
J.-S. Guo and P. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., 331 (2005), 651-667.
doi: 10.1007/s00208-004-0601-7. |
[14] |
J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J., 60 (2008), 37-70. |
[15] |
Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Theory, Kluwer Academic Publishers, Dordrecht, 1997.
doi: 10.1007/978-1-4615-6359-4. |
[16] |
Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, Applications, Kluwer Academic Publishers, Dordrecht, 2000.
doi: 10.1007/978-1-4615-4665-8_17. |
[17] |
A. G. Ibrahim, On differential inclusions with memory in Banach spaces, Proc. Math. Phys. Soc. Egypt, 67 (1992), 1-26. |
[18] |
A. G. Ibrahim, Topological properties of solution sets for functional differential inclusions governed by a family of operators, Portugal. Math., 58 (2001), 255-270. |
[19] |
O. V. Kapustyan, V. S. Mel'nik, J. Valero and V. V. Yasinsky, Global Attractors of Multi-valued Dynamical Systems and Evolution Equations without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, Kyiv, 2008. |
[20] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, 1968. |
[21] |
Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. |
[22] |
N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations, Lecture Notes in Mathematics, 1260, Springer, Berlin, 1987. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007. |
[25] |
H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988. |
[26] |
G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002. |
[27] |
A. A. Tolstonogov, Solutions of evolution inclusions. I, Siberian Math. J., 33 (1993), 500-511.
doi: 10.1007/BF00970899. |
[28] |
A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II, Siberian Math. J., 33 (1993), 693-702.
doi: 10.1007/BF00971135. |
[29] |
A. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, 2000.
doi: 10.1007/978-94-015-9490-5. |
[30] |
W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Spaces, second edition, Elsevier, Amsterdam, 2003. |
[2] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. |
[3] |
J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory, Springer, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[4] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Noordhoff International Publishing, 1976. |
[5] |
Ph. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972), A47-A50. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[7] |
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. |
[8] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, 1977. |
[9] |
Xinfu Chen, J.-S. Guo and Bei Hu, Dead-core rates for the porous medium equation with a strong absorption, Discrete and Continuous Dynamical Systems, Series B, to appear.
doi: 10.3934/dcdsb.2012.17.1761. |
[10] |
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511566158. |
[11] |
J. Diestel and Jr. J. J. Uhl, Vector Measures, American Mathematical Society, Providence, 1977. |
[12] |
A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math., 57 (2000), 203-217. |
[13] |
J.-S. Guo and P. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., 331 (2005), 651-667.
doi: 10.1007/s00208-004-0601-7. |
[14] |
J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J., 60 (2008), 37-70. |
[15] |
Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Theory, Kluwer Academic Publishers, Dordrecht, 1997.
doi: 10.1007/978-1-4615-6359-4. |
[16] |
Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, Applications, Kluwer Academic Publishers, Dordrecht, 2000.
doi: 10.1007/978-1-4615-4665-8_17. |
[17] |
A. G. Ibrahim, On differential inclusions with memory in Banach spaces, Proc. Math. Phys. Soc. Egypt, 67 (1992), 1-26. |
[18] |
A. G. Ibrahim, Topological properties of solution sets for functional differential inclusions governed by a family of operators, Portugal. Math., 58 (2001), 255-270. |
[19] |
O. V. Kapustyan, V. S. Mel'nik, J. Valero and V. V. Yasinsky, Global Attractors of Multi-valued Dynamical Systems and Evolution Equations without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, Kyiv, 2008. |
[20] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, 1968. |
[21] |
Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. |
[22] |
N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations, Lecture Notes in Mathematics, 1260, Springer, Berlin, 1987. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007. |
[25] |
H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988. |
[26] |
G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002. |
[27] |
A. A. Tolstonogov, Solutions of evolution inclusions. I, Siberian Math. J., 33 (1993), 500-511.
doi: 10.1007/BF00970899. |
[28] |
A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II, Siberian Math. J., 33 (1993), 693-702.
doi: 10.1007/BF00971135. |
[29] |
A. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, 2000.
doi: 10.1007/978-94-015-9490-5. |
[30] |
W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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