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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

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 & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907
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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). Google Scholar

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Xinfu Chen, J.-S. Guo and Bei Hu, Dead-core rates for the porous medium equation with a strong absorption,, \emph{Discrete and Continuous Dynamical Systems, (). doi: 10.3934/dcdsb.2012.17.1761. Google Scholar

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A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces,, \emph{Portugal. Math.}, 57 (2000), 203. Google Scholar

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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,, \emph{Math. Ann.}, 331 (2005), 651. doi: 10.1007/s00208-004-0601-7. Google Scholar

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J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption,, \emph{Tohoku Math. J.}, 60 (2008), 37. Google Scholar

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Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Theory,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-1-4615-6359-4. Google Scholar

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Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, Applications,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4615-4665-8_17. Google Scholar

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A. G. Ibrahim, On differential inclusions with memory in Banach spaces,, \emph{Proc. Math. Phys. Soc. Egypt}, 67 (1992), 1. Google Scholar

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A. G. Ibrahim, Topological properties of solution sets for functional differential inclusions governed by a family of operators,, \emph{Portugal. Math.}, 58 (2001), 255. Google Scholar

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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, (2008). Google Scholar

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O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type,, Translations of Mathematical Monographs 23, (1968). Google Scholar

[21]

Ch. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer, (1966). Google Scholar

[22]

N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations,, Lecture Notes in Mathematics, (1260). Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkh\, (2007). Google Scholar

[25]

H. L. Royden, Real Analysis,, third edition, (1988). Google Scholar

[26]

G. V. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002). Google Scholar

[27]

A. A. Tolstonogov, Solutions of evolution inclusions. I,, \emph{Siberian Math. J.}, 33 (1993), 500. doi: 10.1007/BF00970899. Google Scholar

[28]

A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II,, \emph{ Siberian Math. J.}, 33 (1993), 693. doi: 10.1007/BF00971135. Google Scholar

[29]

A. A. Tolstonogov, Differential Inclusions in a Banach Space,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-94-015-9490-5. Google Scholar

[30]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces,, second edition, (2003). Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkh\, (1990). Google Scholar

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory,, Springer, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar

[4]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste Rom\^ania, (1976). Google Scholar

[5]

Ph. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach,, \emph{C. R. Acad. Sci. Paris S\'er. A-B}, 274 (1972). Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar

[7]

C. Castaing, L. A. Faik and A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay,, \emph{Int. J. Appl. Math.}, 2 (2000), 1005. Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). Google Scholar

[9]

Xinfu Chen, J.-S. Guo and Bei Hu, Dead-core rates for the porous medium equation with a strong absorption,, \emph{Discrete and Continuous Dynamical Systems, (). doi: 10.3934/dcdsb.2012.17.1761. Google Scholar

[10]

E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1989). doi: 10.1017/CBO9780511566158. Google Scholar

[11]

J. Diestel and Jr. J. J. Uhl, Vector Measures,, American Mathematical Society, (1977). Google Scholar

[12]

A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces,, \emph{Portugal. Math.}, 57 (2000), 203. Google Scholar

[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,, \emph{Math. Ann.}, 331 (2005), 651. doi: 10.1007/s00208-004-0601-7. Google Scholar

[14]

J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption,, \emph{Tohoku Math. J.}, 60 (2008), 37. Google Scholar

[15]

Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Theory,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-1-4615-6359-4. Google Scholar

[16]

Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, Applications,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4615-4665-8_17. Google Scholar

[17]

A. G. Ibrahim, On differential inclusions with memory in Banach spaces,, \emph{Proc. Math. Phys. Soc. Egypt}, 67 (1992), 1. Google Scholar

[18]

A. G. Ibrahim, Topological properties of solution sets for functional differential inclusions governed by a family of operators,, \emph{Portugal. Math.}, 58 (2001), 255. Google Scholar

[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, (2008). Google Scholar

[20]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type,, Translations of Mathematical Monographs 23, (1968). Google Scholar

[21]

Ch. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer, (1966). Google Scholar

[22]

N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations,, Lecture Notes in Mathematics, (1260). Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkh\, (2007). Google Scholar

[25]

H. L. Royden, Real Analysis,, third edition, (1988). Google Scholar

[26]

G. V. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002). Google Scholar

[27]

A. A. Tolstonogov, Solutions of evolution inclusions. I,, \emph{Siberian Math. J.}, 33 (1993), 500. doi: 10.1007/BF00970899. Google Scholar

[28]

A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II,, \emph{ Siberian Math. J.}, 33 (1993), 693. doi: 10.1007/BF00971135. Google Scholar

[29]

A. A. Tolstonogov, Differential Inclusions in a Banach Space,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-94-015-9490-5. Google Scholar

[30]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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