# American Institute of Mathematical Sciences

March  2012, 32(3): 847-865. doi: 10.3934/dcds.2012.32.847

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

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578 2 Faculty of Symbiotic Systems Science, Fukushima University, Kanayagawa, Fukushima 960-1269, Japan

Received  September 2010 Revised  February 2011 Published  October 2011

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 $u$.
Citation: Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847
##### References:
 [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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##### References:
 [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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