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August  2014, 7(4): 767-783. doi: 10.3934/dcdss.2014.7.767

Global solutions for a nonlinear integral equation with a generalized heat kernel

Kazuhiro Ishige 1, , Tatsuki Kawakami 2, and  Kanako Kobayashi 1,

1. 

Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
2. 

Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan

Received  September 2013 Published  February 2014

We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
Keywords: generalized heat kernel, Global solutions, nonlinear integral equation., weak $L^r$ space, semilinear parabolic equations.
Mathematics Subject Classification: Primary: 35A01, 35K55; Secondary: 35K9.
Citation: Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767
References:
[1]

L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal., 31 (1998), 621-628. doi: 10.1016/S0362-546X(97)00427-6.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar

[4]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. doi: 10.1137/S0036139996313447.  Google Scholar

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S. Benachour, G. Karch and P. Laurençot, Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 1275-1308. doi: 10.1016/j.matpur.2004.03.002.  Google Scholar

[6]

G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3.  Google Scholar

[7]

S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323. doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar

[8]

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Math. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5.  Google Scholar

[9]

M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal., 11 (2012), 1285-1301. doi: 10.3934/cpaa.2012.11.1285.  Google Scholar

[10]

A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384. doi: 10.1007/s00605-009-0093-3.  Google Scholar

[11]

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.  Google Scholar

[12]

V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131.  Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008.  Google Scholar

[14]

F. Gazzola and H.-C. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var. Partial Differential Equations, 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7.  Google Scholar

[15]

K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-525. doi: 10.3792/pja/1195519254.  Google Scholar

[16]

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), 2673-2708. doi: 10.1512/iumj.2009.58.3771.  Google Scholar

[17]

K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math., 121 (2013), 317-351. doi: 10.1007/s11854-013-0038-6.  Google Scholar

[18]

K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().   Google Scholar

[19]

T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.  Google Scholar

[20]

K. Kobayashi, T. Sirao and H. Tanaka, On the glowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.  Google Scholar

[21]

P. Laurençot and P. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math., 89 (2003), 367-383. doi: 10.1007/BF02893088.  Google Scholar

[22]

T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar

[23]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[24]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975.  Google Scholar

[26]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar

[27]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[28]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar

[29]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal., 31 (1998), 621-628. doi: 10.1016/S0362-546X(97)00427-6.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar

[4]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. doi: 10.1137/S0036139996313447.  Google Scholar

[5]

S. Benachour, G. Karch and P. Laurençot, Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 1275-1308. doi: 10.1016/j.matpur.2004.03.002.  Google Scholar

[6]

G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3.  Google Scholar

[7]

S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323. doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar

[8]

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Math. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5.  Google Scholar

[9]

M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal., 11 (2012), 1285-1301. doi: 10.3934/cpaa.2012.11.1285.  Google Scholar

[10]

A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384. doi: 10.1007/s00605-009-0093-3.  Google Scholar

[11]

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.  Google Scholar

[12]

V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131.  Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008.  Google Scholar

[14]

F. Gazzola and H.-C. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var. Partial Differential Equations, 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7.  Google Scholar

[15]

K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-525. doi: 10.3792/pja/1195519254.  Google Scholar

[16]

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), 2673-2708. doi: 10.1512/iumj.2009.58.3771.  Google Scholar

[17]

K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math., 121 (2013), 317-351. doi: 10.1007/s11854-013-0038-6.  Google Scholar

[18]

K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().   Google Scholar

[19]

T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.  Google Scholar

[20]

K. Kobayashi, T. Sirao and H. Tanaka, On the glowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.  Google Scholar

[21]

P. Laurençot and P. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math., 89 (2003), 367-383. doi: 10.1007/BF02893088.  Google Scholar

[22]

T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar

[23]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[24]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975.  Google Scholar

[26]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar

[27]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[28]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar

[29]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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