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Global solutions for a semilinear heat equation in the exterior domain of a compact set

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  • Let $u$ be a global in time solution of the Cauchy-Dirichlet problem for a semilinear heat equation, $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u+u^p,\quad & x\in\Omega,\,\, t>0,\\ u=0,\quad & x\in\partial\Omega,\,\,t>0,\\ u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega, \end{array} \right. $$ where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$, $\Omega$ is a smooth domain in ${\bf R}^N$, and $\phi\in L^\infty(\Omega)$. In this paper we give a sufficient condition for the solution $u$ to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$, and give a classification of the large time behavior of the solution $u$.
    Mathematics Subject Classification: Primary: 35B40, 35K58; Secondary: 35B44.

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  • [1]

    C. Bandle and H. A. Levine, Fujita type results for convective-like reaction diffusion equations in exterior domains, Z. Angew. Math. Phys., 40 (1989), 665-676.

    [2]

    M.-F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.doi: 10.1007/BF00251552.

    [3]

    M.-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.doi: 10.1007/BF02788105.

    [4]

    T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations, 9 (1984), 955-978.

    [5]

    M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133.doi: 10.1016/0362-546X(87)90001-0.

    [6]

    A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbfR^N$, J. Math. Pures. Appl., 87 (2007), 537-561.doi: 10.1016/j.matpur.2007.03.001.

    [7]

    H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.

    [8]

    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.doi: 10.1002/cpa.3160340406.

    [9]

    B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.

    [10]

    Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421.doi: 10.1007/BF01211756.

    [11]

    A. Grigor'yan and L. Saloff-Coste, Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math., 55 (2002), 93-133.doi: 10.1002/cpa.10014.

    [12]

    K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J., 155 (1999), 1-26.

    [13]

    K. Ishige, An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Differential Equations, 158 (1999), 251-290.doi: 10.1006/jdeq.1999.3646.

    [14]

    K. Ishige, Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition, Adv. Differential Equations, 12 (2007), 1135-1166.

    [15]

    K. Ishige, M. Ishiwata and T. Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J., 58 (2009), 2673-2707.doi: 10.1512/iumj.2009.58.3771.

    [16]

    K. Ishige and T. Kawakami, Global solutions of the heat equation with a nonlinear boundary condition, Calc. Var. Partial Differential Equations, 39 (2010), 429-457.

    [17]

    O. Kavian, Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.

    [18]

    T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1-15.

    [19]

    O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," (Russian), Izdat. "Nauka," Moscow, 1968.

    [20]

    M. Murata, Nonuniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Funct. Anal., 135 (1996), 456-487.doi: 10.1006/jfan.1996.0016.

    [21]

    R. Pinsky, The Fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain, J. Differential Equations, 246 (2009), 2561-2576.doi: 10.1016/j.jde.2008.07.029.

    [22]

    S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.

    [23]

    P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dyn. Syst., 21 (2008), 307-318.doi: 10.3934/dcds.2008.21.307.

    [24]

    P. Quittner and P. Souplet, "Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.

    [25]

    S. Salsa, Some properties of nonnegative solutions of parabolic differential operators, Ann. Mat. Pura Appl., 128 (1981), 193-206.doi: 10.1007/BF01789473.

    [26]

    K. Takaichi, Boundedness of global solutions for some semilinear parabolic problems on general domains, Adv. Math. Sci. Appl., 16 (2006), 479-490.

    [27]

    M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996.

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