Research activity

My main research interests can be divided, in broad terms, according to the following topics (which are however related to one another):

• Nonlinear diffusion with weights: see papers [1, 2, 3, 4, 11, 14, 32] and preprint [36].

• Nonlinear diffusion on Riemannian manifolds: see papers [5, 8, 13, 16, 17, 18, 19, 20, 23, 24, 26, 30] and preprint [34]

• Functional inequalities in weighted and Riemannian settings: see papers [10, 12, 15, 25, 27, 28].

• Fractional diffusion and related functional calculus: see papers [6, 7, 9, 22, 33].

• Nonlinear elliptic PDEs on Riemannian manifolds: see paper [31] and preprint [35].

• Gradient flows in non-Euclidean settings: see papers [21, 29].

Papers in refereed journals

1) 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. 33 (2013), 3599–3640. Preprint arXiv.

2) G. Grillo and M. Muratori, Sharp short and long time L-infinity bounds for solutions to porous media equations with homogeneous Neumann boundary conditions, J. Differential Equations 254 (2013), 2261–2288. Preprint arXiv.

3) G. Grillo, M. Muratori and F. Punzo, Conditions at infinity for the inhomogeneous filtration equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 413–428. Preprint arXiv.

4) G. Grillo and M. Muratori, Sharp asymptotics for the porous media equation in low dimensions via Gagliardo-Nirenberg inequalities, Riv. Math. Univ. Parma (N.S.) 5 (2014), 15–38.

5) G. Grillo and M. Muratori, Radial fast diffusion on the hyperbolic space, Proc. Lond. Math. Soc. 109 (2014), 283–317. Preprint arXiv.

6) G. Grillo, M. Muratori and F. Punzo, On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst. 35 (2015), 5927–5962. Preprint arXiv.

7) G. Grillo, M. Muratori and F. Punzo, Fractional porous media equations: existence and uniqueness of weak solutions with measure data, Calc. Var. Partial Differential Equations 54 (2015), 3303–3335. Preprint arXiv.

8) G. Grillo and M. Muratori, Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds, Nonlinear Anal. 131 (2016), 346–362. Preprint arXiv.

9) M. Muratori, The fractional Laplacian in power-weighted Lp spaces: Integration-by-parts formulas and self-adjointness, J. Funct. Anal. 271 (2016), 3662–3694. Preprint arXiv.

10) M. Bonforte, J. Dolbeault, M. Muratori and B. Nazaret, Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities, Kinet. Relat. Models 10 (2017), 33–59. Preprint arXiv.

11) M. Bonforte, J. Dolbeault, M. Muratori and B. Nazaret, Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods, Kinet. Relat. Models 10 (2017), 61–91. Preprint arXiv.

12) J. Dolbeault, M. J. Esteban, M. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities, C. R. Math. Acad. Sci. Paris 355 (2017), 133–154. Preprint arXiv.

13) G. Grillo, M. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour, Adv. Math. 314 (2017), 328–377. Preprint arXiv.

14) A. R. Fotache and M. Muratori, Smoothing effects for the filtration equation with different powers, J. Differential Equations 263 (2017), 3291–3326. Preprint arXiv.

15) J. Dolbeault, M. Muratori and B. Nazaret, Weighted interpolation inequalities: a perturbation approach, Math. Ann. 369 (2017), 1237–1270. Preprint arXiv.

16) G. Grillo, M. Muratori and F. Punzo, The porous medium equation with large initial data on negatively curved Riemannian manifolds, J. Math. Pures Appl. 113 (2018), 195–226. Preprint arXiv.

17) G. Grillo, M. Muratori and F. Punzo, The porous medium equation with measure data on negatively curved Riemannian manifolds, J. Eur. Math. Soc. (JEMS) 20 (2018), 2769–2812. Preprint arXiv.

18) G. Grillo, M. Muratori and F. Punzo, Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds, J. Differential Equations 266 (2019), 4305–4336. Preprint arXiv.

19) M. Muratori and F. Punzo, Porous medium equations on manifolds with critical negative curvature: unbounded initial data, Appl. Anal. 98 (2019), 1756–1772. Preprint arXiv.

20) G. Grillo, M. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case, Math. Ann. 373 (2019), 119–153. Preprint arXiv.

21) M. Muratori and G. Savaré, Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties, J. Funct. Anal. 278 (2020), 108347, 67 pp. Preprint arXiv.

22) G. Grillo, M. Muratori and F. Punzo, Uniqueness of very weak solutions for a fractional filtration equation, Adv. Math. 365 (2020), 107041, 35 pp. Preprint arXiv.

23) G. Grillo, K. Ishige and M. Muratori, Nonlinear characterizations of stochastic completeness, J. Math. Pures Appl. 139 (2020), 63–82. Preprint arXiv.

24) M. Muratori, Some recent advances in nonlinear diffusion on negatively-curved Riemannian manifolds: from barriers to smoothing effects (Survey Paper), Boll. Unione Mat. Ital. 14 (2021), 69–97.

25) T. Kawakami and M. Muratori, Nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds, Geometric Properties for Parabolic and Elliptic PDE’s, 183–203, Springer INdAM Ser., 47, Springer, Cham (2021). Preprint arXiv.

26) G. Grillo, M. Muratori and F. Punzo, Fast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations, Trans. Amer. Math. Soc. 374 (2021), 6367–6396. Preprint arXiv.

27) M. Muratori and A. Roncoroni, Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusion equations, Potential Anal. 57 (2022), 129–154. Preprint arXiv.

28) D. Addona, M. Muratori and M. Rossi, On the equivalence of Sobolev norms in Malliavin spaces, J. Funct. Anal. 283 (2022), 109600, 41 pp. Preprint arXiv.

29) N. De Ponti, M. Muratori and C. Orrieri, Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below, J. Funct. Anal. 283 (2022), 109661, 58 pp. Preprint arXiv.

30) G. Grillo, M. Muratori and F. Punzo, The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds, Discrete Contin. Dyn. Syst. 43 (2023), 1469–1498. Preprint arXiv.

31) M. Muratori and N. Soave, Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (2023), 751–792. Preprint arXiv.

32) M. Muratori and T. Petitt, An inhomogeneous porous medium equation with large data: well-posedness, J. Differential Equations 377 (2023), 712–758. Preprint arXiv.

33) E. Berchio, M. Bonforte, G. Grillo and M. Muratori, The Fractional Porous Medium Equation on noncompact Riemannian manifolds, in press in "Mathematische Annalen", DOI: 10.1007/s00208-023-02731-6. Preprint arXiv.

Preprints

34) G. Grillo, K. Ishige, M. Muratori and F. Punzo, A general nonlinear characterization of stochastic incompleteness, submitted, preprint arXiv.

35) M. Muratori and N. Soave, The Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity of radial solutions, submitted, preprint arXiv.

36) M. Muratori, T. Petitt and F. Quirós,  An inhomogeneous porous medium equation with non-integrable data: asymptotics, submitted, preprint arXiv.