Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Gabriele Grillo [1] and Matteo Muratori [1]

Sharp asymptotics for the porous media equation in low dimensions via Gagliardo-Nirenberg inequalities

Pages: 15-38
Accepted: 30 April 2013
Mathematics Subject Classification (2010): Primary: 35K55, 35B40; Secondary: 35K65, 39B62.

Keywords: Weighted porous media equation, smoothing effect, asymptotic behaviour, weighted Poincaré, Sobolev and Gagliardo-Nirenberg inequalities, nonlinear diffusion equations.
[1] : Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Abstract: We prove sharp asymptotic bounds for solutions to the porous media equation with homogeneous Dirichlet or Neumann boundary conditions on a bounded Euclidean domain, in dimension $$N=1$$ and $$N=2$$. This is achieved by making use of appropriate Gagliardo-Nirenberg inequalities only. The generality of the discussion allows to prove similar bounds for weighted porous media equations, provided one deals with weights for which suitable Gagliardo-Nirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions.

References

[1] N. D. Alikakos, $$L^p$$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827-868.
[2] N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J. 30 (1981), 749-785.
[3] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
[4] M. Bonforte and G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities, J. Funct. Anal. 225 (2005), 33-62.
[5] S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1- dimensional Poincaré inequalities, Indiana Univ. Math. J. 49 (2000), 143- 175.
[6] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259-275.
[7] J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, $$L^q$$-functional inequalities and weighted porous media equations, Potential Anal. 28 (2008), 35-59.
[8] J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci. 6 (2008), 477-494.
[9] D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations 84 (1990), 309-318.
[10] D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc. 120 (1994), 825-830.
[11] G. Grillo and M. Muratori, Sharp short and long time $$L^\infty$$ bounds for solutions to porous media equations with homogeneous Neumann boundary conditions, J. Differential Equations 254 (2013), 2261-2288.
[12] G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete Contin. Dyn. Syst. A 33 (2013), 3599-3640.
[13] G. Grillo, M. Muratori and F. Punzo, Conditions at infinity for the inhomogeneous filtration equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, 413-428.
[14] S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst. 26 (2010), 521-549.
[15] S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math. 34 (1981), 831-852.
[16] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math. 35 (1982), 113-127.
[17] B. Opic and A. Kufner, Hardy-type inequalities, Longman Scientific & Technical, Harlow 1990.
[18] A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math. 65 (2012), 1242-1284.
[19] M. A. Peletier, A supersolution for the porous media equation with nonuniform density, Appl. Math. Lett. 7 (1994), 29-32.
[20] F. Punzo, Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds, Asymptot. Anal. 79 (2012), 273-301.
[21] G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media 1 (2006), 337-351.
[22] G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal. 7 (2008), 1275-1294.
[23] G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal. 8 (2009), 493-508.
[24] J. L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type, Oxford University Press, Ox- ford 2006.
[25] J. L. Vázquez, The porous medium equation. Mathematical theory, The Clarendon Press, Oxford University Press, Oxford 2007.

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