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

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

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  • 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.

    Mathematics Subject Classification: Primary: 35B40, 35K55, 35K92.

    Citation:

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  • [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.
    [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.
    [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.
    [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.
    [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.
    [6] J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear.
    [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.
    [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.
    [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. 
    [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.
    [11] S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.  doi: 10.1007/BF02761536.
    [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. 
    [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.
    [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.
    [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.
    [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.
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