Fabio Punzo

Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds

We study, on weighted Riemannian model manifolds, well posedness of the Cauchy problem for a class of quasilinear parabolic equations with a coefficient which can be singular at infinity. We establish uniqueness or non-uniqueness of bounded solutions, under suitable assumptions on the behavior at infinity of the singular coefficient and on the Green function for the weighted Laplace-Beltrami operator.

On a semilinear parabolic equation with inverse-square potential

We study existence and uniqueness, nonexistence and nonuniqueness of nonnegative solutions to a semilinear parabolic equation with inverse-square potential. Analogous existence and nonexistence results for the companion elliptic equation were proved in [4]. Concerning nonuniqueness, we extend the results proved in [16] for the case without potential.

Prescribed conditions at infinity for parabolic equations

We are concerned with existence and uniqueness of the solutions for linear and nonlinear parabolic equations with time-dependent coefficients, in the class of bounded solutions satisfying appropriate conditions at infinity.

On a fractional sublinear elliptic equation with a variable coefficient

We study the existence and uniqueness of bounded weak solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the proof of uniqueness relies on uniqueness of weak solutions to an associated fractional porous medium equation with variable density.

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

ABSTRACT We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan–Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.

Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakkes motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakkes motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huiskens monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.