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Global solutions for a nonlinear integral equation with a generalized heat kernel
1. | Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan |
2. | Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008. |
[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. |
[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. |
[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. |
[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. |
[18] |
K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().
|
[19] |
T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400. |
[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. |
[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. |
[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. |
[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. |
[24] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007. |
[25] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975. |
[26] |
S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51. |
[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. |
[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. |
[29] |
W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008. |
[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. |
[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. |
[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. |
[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. |
[18] |
K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().
|
[19] |
T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400. |
[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. |
[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. |
[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. |
[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. |
[24] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007. |
[25] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975. |
[26] |
S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51. |
[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. |
[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. |
[29] |
W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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