Reaction-diffusion equations with a switched--off reaction zone

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September 2014, 13(5): 1907-1933. doi: 10.3934/cpaa.2014.13.1907

Reaction-diffusion equations with a switched--off reaction zone

Peter E. Kloeden ^1, , Thomas Lorenz ^2, and Meihua Yang ^3,

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

Received March 2013 Revised May 2013 Published June 2014

Abstract
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Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.

Keywords: Reaction-diffusion equation, dead core, existence of solutions., memory, inclusion equations, discontinuous right-hand sides.

Mathematics Subject Classification: Primary: 35R70; Secondary: 35K15, 35K5.

Citation: Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

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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