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

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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.
    Mathematics Subject Classification: Primary: 35R70; Secondary: 35K15, 35K57.

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  • [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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