**Gabriele Grillo** ^{[1]} and **Matteo Muratori** ^{[1]}

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

**Pages:** 15-38

**Received:** 25 March 2013

**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.

**Authors addresses:**

[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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