Research

My research interest are linear and non-linear parabolic problems and elliptic equations posed on Riemannian manifolds and infinite Graphs. I have addressed questions of global existence in time and blow-up of solutions as well as uniqueness of solutions. I have also worked on variational formulation of solutions in terms of gradient flow to evolutive non-linear PDEs.

Publications:

1. G. Grillo, G. Meglioli, F. Punzo, Blow-up and global existence for semilinear parabolic equations on infinite graphs, Calc Var. and PDEs, accepted (2026)

2. A. Grigoryan, G. Meglioli, A. Roncoroni, Rigidity for the heat equation with density on Riemannian manifolds through a conformal change, Proceedings of the Royal Society of Edinburgh: Section A Mathematics,  1--41 (2026).

3. G. Meglioli, F. Oliva, F. Petitta, Global existence for a Leibenson type equation with reaction on Riemannian manifolds, Nonlin. Anal., 263, 113967, (2026). 

4. G. Meglioli, On the uniqueness for the heat equation with density on infinite graphs, J. Differential Equations, 425, 728--762, (2025).

5. S. Biagi, G. Meglioli, F. Punzo, Uniqueness for local-nonlocal elliptic equations, Comm. Contemporary Mathematics, 2550017, (2025).

6. G. Meglioli, F. Punzo, Uniqueness in weighted l^p spaces for the Schr\"odinger equation on infinite graphs, Proc. Amer. Math. Soc., 153(4), 1519--1537 (2025).

7. S. Biagi, G. Meglioli, F. Punzo, A Liouville theorem for  elliptic equations with a potential on infinite graphs, Calc. Var. and PDEs, 63(7), 165, (2024).

8. G. Meglioli, A. Roncoroni, Uniqueness in Weighted Lebesgue Spaces for an Elliptic Equation with Drift on Manifolds, J Geom. Anal. 33 320 (2023).

9. G. Meglioli, F. Punzo, Uniqueness for fractional parabolic and elliptic equations with drift, Comm. Pure Appl. Math. 22(6), 1962-1981, (2023).

10. G. Meglioli, Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients, Discr. Cont. Dynam. System - Series A, 43(6), 2305-2336, (2023).

11. G. Grillo, G. Meglioli, F. Punzo Global existence for reaction-diffusion evolution equations driven by the p-Laplacian on manifolds, Mathematics in Engineering 5(3) (2023) 1-38.

12. G. Grillo, G. Meglioli, F. Punzo Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds, Annali di Matematica Pura ed Applicata, 202, 1255-1270, (2022).

13. G. Meglioli, D.D. Monticelli, F. Punzo, Nonexistence of solutions to quasilinear parabolic equations with a potential, Calc. Var. and PDEs 61, 23 (2022).

14. G. Grillo, G. Meglioli, F. Punzo, Global existence of solutions and smoothing effects for a class of reaction-diffusion equations on manifolds, J. Evolution Equations, 21(2), 2339-2375, (2021).

15. G. Meglioli, F. Punzo, Blow-up and global existence for the inhomogeneous porous medium equation with reaction, Rend. Mat. Appl., 42, 271-292, (2021).

16. G. Grillo, G. Meglioli, F. Punzo, Smoothing effects and infinite time blowup for reaction-diffusion equations on manifolds: an approach via Sobolev and Poincaré inequalities, J. Math. Pures et Appl., 151, 99-131, (2021).

17. G. Meglioli, F. Punzo, Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density, Nonlin. Anal., 203, 112-187, (2021).

18. G. Meglioli, F. Punzo, Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density, J. Diff. Eq. 269, 8918-8958, (2020).

Preprints:

1. M. Erbar, G. Meglioli, Gradient flow for a class of diffusion equations with Dirichlet boundary data, submitted (2024).

2. G. Meglioli, F. Punzo, Uniqueness of solutions to elliptic and parabolic equations on metric graphs, submitted (2025).

3. S. Biagi, G. Meglioli, F. Punzo, Phragmen-Lindelof type theorems for parabolic equations on infinite graphs, submitted (2025).