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Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity
Large time behavior of ODE type solutions to nonlinear diffusion equations
1. | Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan |
2. | Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan |
$ \begin{equation} \left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} $ |
$ m>0 $ |
$ \alpha\in(-\infty,1) $ |
$ \lambda>0 $ |
$ \varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N) $ |
$ 1\le r<\infty $ |
$ \inf_{x\in{\bf R}^N}\varphi(x)>-\lambda $ |
$ \zeta' = \zeta^\alpha $ |
$ (0,\infty) $ |
$ +\infty $ |
$ t\to\infty $ |
References:
[1] |
J. Aguirre and M. A. Escobedo, Cauchy problem for $u_t - \Delta u = u^p$ with $0 < p < 1$, Asymptotic behaviour of solutions, Ann. Fac, Sci. Toulouse Math., 8 (1986/87), 175-203. |
[2] |
D. G. Aronson,
Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.
|
[3] |
J. A. Carrillo and G. Toscani,
Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142.
doi: 10.1512/iumj.2000.49.1756. |
[4] |
A. Friedman and S. Kamin,
The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.
doi: 10.2307/1999846. |
[5] |
S. Kamin and L. A. Peletier,
Source-type solutions of degenerate diffusion equations with absorption, Israel J. Math., 50 (1985), 219-230.
doi: 10.1007/BF02761403. |
[6] |
S. Kamin and L. A. Peletier,
Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146.
doi: 10.1007/BF02801989. |
[7] |
K. Ishige, M. Ishiwata and 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. |
[8] |
K. Ishige and T. Kawakami,
Refined asymptotic profiles for a semilinear heat equation, Math. Ann., 353 (2012), 161-192.
doi: 10.1007/s00208-011-0677-9. |
[9] |
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. |
[10] |
K. Ishige, T. Kawakami and K. Kobayashi,
Asymptotics for a nonlinear integral equation with a generalized heat kernel, J. Evol. Eqn., 14 (2014), 749-777.
doi: 10.1007/s00028-014-0237-3. |
[11] |
K. Ishige and K. Kobayashi,
Convection-diffusion equation with absorption and non-decaying initial data, J. Differential Equations, 254 (2013), 1247-1268.
doi: 10.1016/j.jde.2012.10.014. |
[12] |
K. Ishige, T. Kawakami and H. Michihisa,
Asymptotic expansions of solutions of fractional diffusion equations, SIAM J. Math. Anal., 49 (2017), 2167-2190.
doi: 10.1137/16M1101428. |
[13] |
R. Kajikiya,
Stability and instability of stationary solutions for sublinear parabolic equations, J. Differential Equations, 264 (2018), 786-834.
doi: 10.1016/j.jde.2017.09.023. |
[14] |
T. Kawanago,
Existence and behaviour of solutions for $u_t = \Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.
|
[15] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968. |
[16] |
L. A. Peletier and J. Zhao,
Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 17 (1991), 991-1009.
doi: 10.1016/0362-546X(91)90059-A. |
[17] |
R. Suzuki,
Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity, J. Differential Equations, 190 (2003), 150-181.
doi: 10.1016/S0022-0396(02)00086-4. |
[18] |
N. Umeda,
Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372.
doi: 10.3836/tjm/1244208595. |
[19] |
J. L. Vázquez,
Asymptotic beahviour for the porous medium equation posed in the whole space, J. Evol. Eqn., 3 (2003), 67-118.
doi: 10.1007/s000280300004. |
[20] |
J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[21] |
L. Wang and J. Yin, Grow-up rate of solutions for the heat equation with a sublinear source, Bound. Value Probl., 96 (2012), 14 pp.
doi: 10.1186/1687-2770-2012-96. |
show all references
References:
[1] |
J. Aguirre and M. A. Escobedo, Cauchy problem for $u_t - \Delta u = u^p$ with $0 < p < 1$, Asymptotic behaviour of solutions, Ann. Fac, Sci. Toulouse Math., 8 (1986/87), 175-203. |
[2] |
D. G. Aronson,
Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.
|
[3] |
J. A. Carrillo and G. Toscani,
Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142.
doi: 10.1512/iumj.2000.49.1756. |
[4] |
A. Friedman and S. Kamin,
The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.
doi: 10.2307/1999846. |
[5] |
S. Kamin and L. A. Peletier,
Source-type solutions of degenerate diffusion equations with absorption, Israel J. Math., 50 (1985), 219-230.
doi: 10.1007/BF02761403. |
[6] |
S. Kamin and L. A. Peletier,
Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146.
doi: 10.1007/BF02801989. |
[7] |
K. Ishige, M. Ishiwata and 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. |
[8] |
K. Ishige and T. Kawakami,
Refined asymptotic profiles for a semilinear heat equation, Math. Ann., 353 (2012), 161-192.
doi: 10.1007/s00208-011-0677-9. |
[9] |
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. |
[10] |
K. Ishige, T. Kawakami and K. Kobayashi,
Asymptotics for a nonlinear integral equation with a generalized heat kernel, J. Evol. Eqn., 14 (2014), 749-777.
doi: 10.1007/s00028-014-0237-3. |
[11] |
K. Ishige and K. Kobayashi,
Convection-diffusion equation with absorption and non-decaying initial data, J. Differential Equations, 254 (2013), 1247-1268.
doi: 10.1016/j.jde.2012.10.014. |
[12] |
K. Ishige, T. Kawakami and H. Michihisa,
Asymptotic expansions of solutions of fractional diffusion equations, SIAM J. Math. Anal., 49 (2017), 2167-2190.
doi: 10.1137/16M1101428. |
[13] |
R. Kajikiya,
Stability and instability of stationary solutions for sublinear parabolic equations, J. Differential Equations, 264 (2018), 786-834.
doi: 10.1016/j.jde.2017.09.023. |
[14] |
T. Kawanago,
Existence and behaviour of solutions for $u_t = \Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.
|
[15] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968. |
[16] |
L. A. Peletier and J. Zhao,
Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 17 (1991), 991-1009.
doi: 10.1016/0362-546X(91)90059-A. |
[17] |
R. Suzuki,
Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity, J. Differential Equations, 190 (2003), 150-181.
doi: 10.1016/S0022-0396(02)00086-4. |
[18] |
N. Umeda,
Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372.
doi: 10.3836/tjm/1244208595. |
[19] |
J. L. Vázquez,
Asymptotic beahviour for the porous medium equation posed in the whole space, J. Evol. Eqn., 3 (2003), 67-118.
doi: 10.1007/s000280300004. |
[20] |
J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[21] |
L. Wang and J. Yin, Grow-up rate of solutions for the heat equation with a sublinear source, Bound. Value Probl., 96 (2012), 14 pp.
doi: 10.1186/1687-2770-2012-96. |
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