# American Institute of Mathematical Sciences

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

## 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

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.
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
##### References:
 [1] R. A. Adams and J. F. Fournier, Sobolev Spaces, second edition, Elsevier, Amsterdam, 2003.  Google Scholar [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar [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.  Google Scholar [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Noordhoff International Publishing, 1976.  Google Scholar [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.  Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar [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.  Google Scholar [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, 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, 92, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar [11] J. Diestel and Jr. J. J. Uhl, Vector Measures, American Mathematical Society, Providence, 1977.  Google Scholar [12] A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math., 57 (2000), 203-217.  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, Math. Ann., 331 (2005), 651-667. 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, Tohoku Math. J., 60 (2008), 37-70.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] A. G. Ibrahim, On differential inclusions with memory in Banach spaces, Proc. Math. Phys. Soc. Egypt, 67 (1992), 1-26.  Google Scholar [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.  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, Naukova Dumka, Kyiv, 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, American Mathematical Society, Providence, 1968. Google Scholar [21] Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.  Google Scholar [22] N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations, Lecture Notes in Mathematics, 1260, Springer, Berlin, 1987.  Google Scholar [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.  Google Scholar [24] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007.  Google Scholar [25] H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar [26] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002.  Google Scholar [27] A. A. Tolstonogov, Solutions of evolution inclusions. I, Siberian Math. J., 33 (1993), 500-511. doi: 10.1007/BF00970899.  Google Scholar [28] A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II, Siberian Math. J., 33 (1993), 693-702. 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, 120, Springer, New York, 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, Elsevier, Amsterdam, 2003.  Google Scholar [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar [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.  Google Scholar [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Noordhoff International Publishing, 1976.  Google Scholar [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.  Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar [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.  Google Scholar [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer, Berlin, 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, 92, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar [11] J. Diestel and Jr. J. J. Uhl, Vector Measures, American Mathematical Society, Providence, 1977.  Google Scholar [12] A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math., 57 (2000), 203-217.  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, Math. Ann., 331 (2005), 651-667. 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, Tohoku Math. J., 60 (2008), 37-70.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] A. G. Ibrahim, On differential inclusions with memory in Banach spaces, Proc. Math. Phys. Soc. Egypt, 67 (1992), 1-26.  Google Scholar [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.  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, Naukova Dumka, Kyiv, 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, American Mathematical Society, Providence, 1968. Google Scholar [21] Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966.  Google Scholar [22] N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations, Lecture Notes in Mathematics, 1260, Springer, Berlin, 1987.  Google Scholar [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.  Google Scholar [24] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007.  Google Scholar [25] H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar [26] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002.  Google Scholar [27] A. A. Tolstonogov, Solutions of evolution inclusions. I, Siberian Math. J., 33 (1993), 500-511. doi: 10.1007/BF00970899.  Google Scholar [28] A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II, Siberian Math. J., 33 (1993), 693-702. 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, 120, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar
 [1] Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587 [2] Angela Alberico, Teresa Alberico, Carlo Sbordone. Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1053-1067. doi: 10.3934/dcds.2011.31.1053 [3] M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 [4] Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397 [5] Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083 [6] Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635 [7] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [8] Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815 [9] Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319 [11] Xiao Wu, Mingkang Ni. Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3249-3266. doi: 10.3934/dcdss.2020341 [12] Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193 [13] Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193 [14] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [15] Chin-Chin Wu, Zhengce Zhang. Dead-core rates for the heat equation with a spatially dependent strong absorption. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2203-2210. doi: 10.3934/dcdsb.2013.18.2203 [16] Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761 [17] Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25 [18] Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209 [19] Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 [20] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

2020 Impact Factor: 1.916

## Metrics

• PDF downloads (47)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]