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The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds
Gabriele Grillo, Matteo Muratori, Fabio Punzo
We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be st…
Fast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations
Gabriele Grillo, Matteo Muratori, Fabio Punzo
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integr…
Nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds
Tatsuki Kawakami, Matteo Muratori
It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension , i.e. any complete, simply connected Riemannian manifold with n…
Uniqueness of very weak solutions for a fractional filtration equation
Gabriele Grillo, Matteo Muratori, Fabio Punzo
We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in . With regards to uniqueness, it was shown…
Nonlinear characterizations of stochastic completeness
Gabriele Grillo, Kazuhiro Ishige, Matteo Muratori
We prove that conservation of probability for the free heat semigroup on a Riemannian manifold (namely stochastic completeness), hence a linear property, is equivalent to uniqu…
Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds
Gabriele Grillo, Matteo Muratori, Fabio Punzo
We consider the porous medium equation with power-type reaction terms on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compa…