arXiv subjects

Gabriele Grillo

Publications and source records attributed to Gabriele Grillo.

At least 19 recordsLinked to original sources

arXiv2025

A porous medium equation with rough weights: sharp Widder theory

We establish an optimal \emph{Widder theory} for a weighted porous medium equation with rough and inhomogeneous density that may be singular at a point and tends to zero at spatial infinity. Specifically, for this equation, we identify a class $X$ of initial measure data that give rise to very weak solutions, we show that non-negative very weak solutions necessarily admit an initial trace in $X$ at time $t=0$, and we prove that any two non-negative solutions having the same initial trace are equal. The corresponding theory for the classical (unweighted) equation was established by exploiting various properties that are not available in our weighted setting, such as the continuity of solutions, the explicit scale invariance of the equation, Aleksandrov's reflection principle, and the Aronson--B\'enilan inequality. Therefore, to complete the Widder theory, we must devise several proofs by means of entirely new methods. We also establish an optimal quantitative \emph{a priori} smoothing estimate for unsigned local solutions without resorting to scale invariance, which seems to be new in this form even for the classical porous medium equation. Finally, we show that non-negative very weak solutions are always locally bounded, and in particular that they have locally finite energy.

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arXiv2025

Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a potential term. Our results reveal the interplay between the manifold's geometry, the power nonlinearity, and the potential's behavior at infinity. Using a test function argument, we identify explicit parameter ranges where nonexistence holds.

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arXiv2024

Blow-up and global existence for semilinear parabolic equations on infinite graphs

We investigate existence of global in time solutions and blow-up of solutions to the semilinear heat equation posed on infinite graphs. The source term is a general function $f(u)$. We always assume that the infimum of the spectrum of the Laplace operator $\lambda_1(G)$ on the graph is positive. According to an interaction between the behavior of $f$ close to $0$ and the value $\lambda_1(G)$, we get the existence of a global in time solution or blow-up of any nonnegative solution, provided that the initial datum is nontrivial.

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arXiv2024

Smoothing effects and extinction in finite time for fractional fast diffusions on Riemannian manifolds

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power that makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual $L^1$ space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted $L^p$ spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates.

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arXiv2024

The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach

We consider the porous medium equation (PME) on complete noncompact manifolds $M$ of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space $X$ of functions, strictly larger than $L^1$, in which the Green function on $M$ appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to $X$. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to $X$ as well.

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arXiv2023

A general nonlinear characterization of stochastic incompleteness

Stochastic incompleteness of a Riemannian manifold $M$ amounts to the nonconservation of probability for the heat semigroup on $M$. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to $\Delta W=\psi(W)$ for one, hence all, general nonlinearity $\psi$ which is only required to be continuous, nondecreasing, with $\psi(0)=0$ and $\psi>0$ in $(0,+\infty)$. Similar statements hold for unsigned (sub)solutions. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation $\partial_t u =\Delta\phi(u)$ with bounded initial data for one, hence all, general nonlinearity $\phi$ which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.

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arXiv2022

Global existence for reaction-diffusion evolution equations driven by the $p$-Laplacian on manifolds

We consider reaction-diffusion equations driven by the $p$-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $L^2$ spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the $L^\infty$ norm of solutions at all positive times, in terms of $L^q$ norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.

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arXiv2022

The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds

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 stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on $[0,T]$ for a suitable $T>0$. The maximal existence time $T$ is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class.

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arXiv2021

Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds

It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $\Lambda$ of the $L^2$ spectrum of $-\Delta$ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $\Lambda$. Such nonlinearities are time-independent, in contrast to the ones studied in [2]. As a consequence of our results we show that, on a class of manifolds much larger than the case $M=\mathbb{H}^N$ considered in [2], solutions to (1.1) with power nonlinearity $f(u)=u^p$, $p>1$, and corresponding to sufficiently small data, are global in time.

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arXiv2021

The Fractional Porous Medium Equation on noncompact Riemannian manifolds

We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual $L^1$ space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold include both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involve a class of data which is larger than previously known one.

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arXiv2021

Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities

We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with pow\-er-type nonlinearity and slow diffusion of porous medium time. We consider the particularly delicate case $p<m$ in problem (1.1), a case largely left open in [21] even when the initial datum is smooth and compactly supported. We prove global existence for L$^m$ data, and a smoothing effect for the evolution, i.e. that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L$^\infty$ norm. As a consequence of this fact and of a result of [21], it follows that on Cartan-Hadamard manifolds with curvature pinched between two strictly negative constants, solutions corresponding to sufficiently large L$^m$ data give rise to solutions that blow up pointwise everywhere in infinite time, a fact that has no Euclidean analogue. The methods of proof of the smoothing effect are functional analytic in character, as they depend solely on the validity of the Sobolev inequality and on the fact that the L$^2$ spectrum of $Δ$ on $M$ is bounded away from zero (namely on the validity of a Poincaré inequality on $M$). As such, they are applicable to different situations, among which we single out the case of (mass) weighted reaction-diffusion equation in the Euclidean setting. In this latter setting, a modification of the methods of [37] allows to deal also, with stronger results for large times, with the case of globally integrable weights.

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arXiv2020

Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds

We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${\mathbb R}^n$.

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arXiv2020

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality cannot be further improved. Such inequalities arise from more general, \emph{optimal} ones valid for the operator $ P_λ:= -Δ_{\mathbb{H}^N} - λ$ where $0 \leq λ\leq λ_{1}(\mathbb{H}^N)$ and $λ_{1}(\mathbb{H}^N)$ is the bottom of the $L^2$ spectrum of $-Δ_{\mathbb{H}^N} $, a problem that had been studied in [J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator $P_{λ_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on $\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact greater generality since there are shown on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_λ.$

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arXiv2020

Fast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations

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 integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{\mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to $-\infty$ at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, under the further assumption that the initial datum is in $L^2_{\mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and we are able to establish uniqueness of purely (nonnegative) distributional solutions, which to our knowledge was not known before even in the Euclidean space. The required curvature bound is in fact sharp, since on model manifolds it turns out to be equivalent to stochastic completeness, and it was shown in [13] that uniqueness for the FDE fails even in the class of bounded solutions on manifolds that are not stochastically complete. Qualitatively this amounts to asking that the curvature diverges at most quadratically at infinity. A crucial ingredient of the uniqueness result is the proof of nonexistence of distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.

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arXiv2020

Nonlinear characterizations of stochastic completeness

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δϕ(u)$, $ϕ$ being an arbitrary concave, increasing positive function, regular outside the origin and with $ϕ(0)=0$. Either property is also shown to be equivalent to nonexistence of nontrivial, nonnegative bounded solutions to the elliptic equation $ΔW=ϕ^{-1}(W)$ with $ϕ$ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds, and on existence or nonexistence of bounded solutions to the mentioned elliptic equations on $M$ are given, these being the first results on such issues.

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arXiv2020

The Fractional Porous Medium Equation on the hyperbolic space

We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $\Delta_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

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arXiv2020

Uniqueness of very weak solutions for a fractional filtration equation

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb R}^d\times(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a consequence, distributional solutions have locally-finite energy.

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arXiv2018

Improved Multipolar Poincaré-Hardy inequalities on Cartan-Hadamard Manifolds

We prove a family of improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincaré inequality such that the critical unipolar singular mass is reached at any pole.

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