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Monotonicity with respect to $p$ of the First Nontrivial Eigenvalue of the $p$-Laplacian with Homogeneous Neumann Boundary Conditions
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) , $ \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 $-Laplacian type equations. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199 ##### References:   E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1.   E. DiBenedetto and A. Friedman, Addendum to Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 363 (1985), 217-220. doi: 10.1515/crll.1985.363.217.   E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.2307/2001442.   E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution$p$-Laplacian equation. Initial traces and Cauchy problem when$1 < p < 2$, Arch. Ration. Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111.   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.   J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. 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.   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.   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.  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.   S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87. doi: 10.1007/BF02761536.   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.  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.   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.   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.   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.   show all references ##### References:   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.   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.   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.   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.   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.   J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. 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.   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.   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.  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.   S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87. doi: 10.1007/BF02761536.   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.  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.   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.   L. A. Peletier and J. N. 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