American Institute of Mathematical Sciences

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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

Kazuhiro Ishige 1, and  Michinori Ishiwata 2,

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 $u$.
Keywords: global solutions., exterior domain, Semilinear heat equation.
Mathematics Subject Classification: Primary: 35B40, 35K58; Secondary: 35B4.
Citation: Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[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. doi: 10.1007/BF00251552. Google Scholar

[3]

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

[4]

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

[5]

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

[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. doi: 10.1016/j.matpur.2007.03.001. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. doi: 10.1002/cpa.10014. Google Scholar

[12]

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

[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. doi: 10.1006/jdeq.1999.3646. Google Scholar

[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. Google Scholar

[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. doi: 10.1512/iumj.2009.58.3771. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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. doi: 10.1016/j.jde.2008.07.029. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). Google Scholar

show all references

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. Google Scholar

[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. doi: 10.1007/BF00251552. Google Scholar

[3]

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

[4]

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

[5]

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

[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. doi: 10.1016/j.matpur.2007.03.001. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. doi: 10.1002/cpa.10014. Google Scholar

[12]

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

[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. doi: 10.1006/jdeq.1999.3646. Google Scholar

[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. Google Scholar

[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. doi: 10.1512/iumj.2009.58.3771. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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. doi: 10.1016/j.jde.2008.07.029. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). Google Scholar

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