# American Institute of Mathematical Sciences

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, (2003). [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkh\, (1990). [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. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste Rom\^ania, (1976). [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). [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). [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. [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). [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. [10] E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1989). doi: 10.1017/CBO9780511566158. [11] J. Diestel and Jr. J. J. Uhl, Vector Measures,, American Mathematical Society, (1977). [12] A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces,, \emph{Portugal. Math.}, 57 (2000), 203. [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. [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. [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. [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. [17] A. G. Ibrahim, On differential inclusions with memory in Banach spaces,, \emph{Proc. Math. Phys. Soc. Egypt}, 67 (1992), 1. [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. [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). [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). [21] Ch. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer, (1966). [22] N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations,, Lecture Notes in Mathematics, (1260). [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. [24] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkh\, (2007). [25] H. L. Royden, Real Analysis,, third edition, (1988). [26] G. V. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002). [27] A. A. Tolstonogov, Solutions of evolution inclusions. I,, \emph{Siberian Math. J.}, 33 (1993), 500. doi: 10.1007/BF00970899. [28] A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II,, \emph{ Siberian Math. J.}, 33 (1993), 693. 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, (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, (2003). [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkh\, (1990). [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. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste Rom\^ania, (1976). [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). [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). [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. [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). [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. [10] E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1989). doi: 10.1017/CBO9780511566158. [11] J. Diestel and Jr. J. J. Uhl, Vector Measures,, American Mathematical Society, (1977). [12] A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces,, \emph{Portugal. Math.}, 57 (2000), 203. [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. [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. [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. [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. [17] A. G. Ibrahim, On differential inclusions with memory in Banach spaces,, \emph{Proc. Math. Phys. Soc. Egypt}, 67 (1992), 1. [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. [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). [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). [21] Ch. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer, (1966). [22] N. Pavel, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations,, Lecture Notes in Mathematics, (1260). [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. [24] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkh\, (2007). [25] H. L. Royden, Real Analysis,, third edition, (1988). [26] G. V. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002). [27] A. A. Tolstonogov, Solutions of evolution inclusions. I,, \emph{Siberian Math. J.}, 33 (1993), 500. doi: 10.1007/BF00970899. [28] A. A. Tolstonogov and Ya. I. Umanskiĭ, Solutions of evolution inclusions. II,, \emph{ Siberian Math. J.}, 33 (1993), 693. 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, (1989). doi: 10.1007/978-1-4612-1015-3.
 [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 - A, 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 - A, 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] 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 [6] 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 [7] 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 [8] 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 [9] Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193 [10] 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 [11] 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 [12] Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209 [13] 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 [14] 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 [15] Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382 [16] 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 - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [17] Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157 [18] Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200 [19] Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 [20] A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

2017 Impact Factor: 0.884