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September  2020, 19(9): 4373-4386. doi: 10.3934/cpaa.2020199

Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

2. 

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

* Corresponding author

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: The second author was partially supported by Grant-in-Aid for Early-Career Scientists JSPS KAKENHI Grant Number 18K13435

Let
$ u $
be a solution to the Cauchy problem for a nonlinear diffusion equation
$ \begin{equation*} \begin{cases} \partial_t u = \mathrm{div}\, (|\nabla u|^{p-2} \nabla u) + u^\alpha & \quad\mathrm{in}\quad{\bf R}^N\times(0, \infty), \\ u(x, 0) = \lambda+\varphi(x) & \quad\mathrm{in}\quad{\bf R}^N, \end{cases} \end{equation*} $
where
$ N \ge 1 $
,
$ 2N/(N+1)<p\neq2 $
,
$ \alpha \in (-\infty, 1) $
,
$ \lambda>0 $
and
$ \varphi\in BC({\bf R}^N)\, \cap\, L^1({\bf R}^N) $
with
$ \varphi\geq0 $
in
$ {\bf R}^{N} $
. Then the solution
$ u $
behaves like a positive solution to ODE
$ \zeta' = \zeta^\alpha $
in
$ (0, \infty) $
. In this paper we show that the large time behavior of the solution
$ u $
is described by a rescaled Barenblatt solution.
Citation: Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199
References:
[1]

E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.  doi: 10.1515/crll.1985.357.1.  Google Scholar

[2]

E. DiBenedetto and A. Friedman, Addendum to Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 363 (1985), 217-220.  doi: 10.1515/crll.1985.363.217.  Google Scholar

[3]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224.  doi: 10.2307/2001442.  Google Scholar

[4]

E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$, Arch. Ration. Mech. Anal., 111 (1990), 225-290.  doi: 10.1007/BF00400111.  Google Scholar

[5]

J. Eom and K. Ishige, Large time behavior of ODE type solutions to a nonlinear parabolic system, Nonlinear Anal., 191 (2020), 19 pp. doi: 10.1016/j.na.2019.111631.  Google Scholar

[6]

J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. Google Scholar

[7]

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.  Google Scholar

[8]

A. Gmira and L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in $\mathbf{R}^{N}$, J. Differ. Equ., 53 (1984), 258-276.  doi: 10.1016/0022-0396(84)90042-1.  Google Scholar

[9]

M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math., 3 (1981), 113-127.   Google Scholar

[10]

K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differ. Equ., 254 (2013), 1247-1268.  doi: 10.1016/j.jde.2012.10.014.  Google Scholar

[11]

S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.  doi: 10.1007/BF02761536.  Google Scholar

[12]

S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12 (1985), 393-408.   Google Scholar

[13]

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.  Google Scholar

[14]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the $p$- Laplacian equation, Rev. Mat. Iberoam., 4 (1988), 339-354.  doi: 10.4171/RMI/77.  Google Scholar

[15]

L. A. Peletier and J. N. Zhao, Source-type solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 14 (1990), 107-121.  doi: 10.1016/0362-546X(90)90018-C.  Google Scholar

[16]

J. N. Zhao, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differ. Equ., 102 (1993), 33-52.  doi: 10.1006/jdeq.1993.1020.  Google Scholar

show all references

References:
[1]

E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.  doi: 10.1515/crll.1985.357.1.  Google Scholar

[2]

E. DiBenedetto and A. Friedman, Addendum to Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 363 (1985), 217-220.  doi: 10.1515/crll.1985.363.217.  Google Scholar

[3]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224.  doi: 10.2307/2001442.  Google Scholar

[4]

E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$, Arch. Ration. Mech. Anal., 111 (1990), 225-290.  doi: 10.1007/BF00400111.  Google Scholar

[5]

J. Eom and K. Ishige, Large time behavior of ODE type solutions to a nonlinear parabolic system, Nonlinear Anal., 191 (2020), 19 pp. doi: 10.1016/j.na.2019.111631.  Google Scholar

[6]

J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. Google Scholar

[7]

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.  Google Scholar

[8]

A. Gmira and L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in $\mathbf{R}^{N}$, J. Differ. Equ., 53 (1984), 258-276.  doi: 10.1016/0022-0396(84)90042-1.  Google Scholar

[9]

M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math., 3 (1981), 113-127.   Google Scholar

[10]

K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differ. Equ., 254 (2013), 1247-1268.  doi: 10.1016/j.jde.2012.10.014.  Google Scholar

[11]

S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.  doi: 10.1007/BF02761536.  Google Scholar

[12]

S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12 (1985), 393-408.   Google Scholar

[13]

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.  Google Scholar

[14]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the $p$- Laplacian equation, Rev. Mat. Iberoam., 4 (1988), 339-354.  doi: 10.4171/RMI/77.  Google Scholar

[15]

L. A. Peletier and J. N. Zhao, Source-type solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 14 (1990), 107-121.  doi: 10.1016/0362-546X(90)90018-C.  Google Scholar

[16]

J. N. Zhao, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differ. Equ., 102 (1993), 33-52.  doi: 10.1006/jdeq.1993.1020.  Google Scholar

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