# Article

MSC: 35K35, 35K59, 47H05
Full entry | PDF   (0.4 MB)
Keywords:
Porous media equations, fast diffusion equations, subdifferential operators
Summary:
This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $$u_t + (-\Delta+1)\beta(u) + G(u) = g \quad \mbox{in}\ \Omega\times(0, T)$$ in an unbounded domain $\Omega \subset \mathbb{R}^N$ with smooth bounded boundary, where $N \in \mathbb{N}$, $T>0$, $\beta$, is a single-valued maximal monotone function on $\mathbb{R}$, e.g., $$\beta(r) = |r|^{q-1}r\ (q > 0, q\neq1)$$ and $G$ is a function on $\mathbb{R}$ which can be regarded as a Lipschitz continuous operator from $(H^1(\Omega))^{*}$ to $(H^1(\Omega))^{*}$. The present work establishes existence and estimates for the above problem.
References:
[1] Akagi, G., Schimperna, G., Segatti, A.: Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations. J. Differential Equations, 261 (2016), pp. 2935–2985. DOI 10.1016/j.jde.2016.05.016 | MR 3527619
[2] Blanchard, D., Porretta, A.: Stefan problems with nonlinear diffusion and convection. J. Differential Equations, 210 (2005), pp. 383–428. DOI 10.1016/j.jde.2004.06.012 | MR 2119989
[3] Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Especes de Hilbert. North-Holland, Amsterdam, 1973. MR 0348562
[4] Damlamian, A.: Some results on the multi-phase Stefan problem. Comm. Partial Differential Equations, 2 (1977), pp. 1017–1044. DOI 10.1080/03605307708820053 | MR 0487015
[5] DiBenedetto, E.: Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J., 32 (1983), pp. 83–118. DOI 10.1512/iumj.1983.32.32008 | MR 0684758
[6] Friedman, A.: The Stefan problem in several space variables. Trans. Amer. Math. Soc., 133 (1968), pp. 51–87. DOI 10.1090/S0002-9947-1968-0227625-7 | MR 0227625
[7] Fukao, T., Kenmochi, N., Pawlow, I.: Transmission problems arising in Czochralski process of crystal growth. Mathematical aspects of modelling structure formation phenomena (Bȩdlewo/Warsaw, 2000), pp. 228–243, GAKUTO Internat. Ser. Math. Sci. Appl., 17, Gakkötosho, Tokyo, 2001. MR 1932116
[8] Fukao, T.: Convergence of Cahn–Hilliard systems to the Stefan problem with dynamic boundary conditions. Asymptot. Anal., 99 (2016), pp. 1–21. DOI 10.3233/ASY-161373 | MR 3541824
[9] Fukao, T., Kurima, S., Yokota, T.: Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy’s criterion. preprint. MR 3790712
[10] Haraux, A., Kenmochi, N.: Asymptotic behaviour of solutions to some degenerate parabolic equations. Funkcial. Ekvac., 34 (1991), pp. 19–38. MR 1116879
[11] Kenig, C. E.: Degenerate Diffusions. Initial value problems and local regularity theory. EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007. MR 2338118
[12] Kurima, S., Yokota, T.: Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates. J. Differential Equations, 263 (2017), pp. 2024–2050. DOI 10.1016/j.jde.2017.03.040 | MR 3650332
[13] Marinoschi, G.: Well-posedness of singular diffusion equations in porous media with homogeneous Neumann boundary conditions. Nonlinear Anal., 72 (2010), pp. 3491–3514. DOI 10.1016/j.na.2009.12.033 | MR 2587381
[14] Okazawa, N., Suzuki, T., Yokota, T.: Energy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory, 1 (2012), pp. 337–354. DOI 10.3934/eect.2012.1.337 | MR 3085232
[15] Rodriguez, A., Vázquez, J. L.: Obstructions to existence in fast-diffusion equations. J. Differential Equations, 184 (2002), pp. 348–385. DOI 10.1006/jdeq.2001.4144 | MR 1929882
[16] Showalter, R. E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. MR 1422252
[17] Vázquez, J. L.: The Porous Medium Equation. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2286292
[18] Yin, H.-M.: On a degenerate parabolic system. J. Differential Equations, 245 (2008), pp. 722–736. DOI 10.1016/j.jde.2008.03.017 | MR 2422525

Partner of