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Fabio Punzo

Publications and source records attributed to Fabio Punzo.

At least 37 records · Page 2Linked to original sources
arXiv2024

Phragmèn-Lindelöf type theorems for elliptic equations on infinite graphs

We investigate the validity of the Phragmèn-Lindelöf principle for a class of elliptic equations with a potential, posed on infinite graphs. Consequently, we get uniqueness, in the class of solutions satisfying a suitable growth condition at infinity. We suppose that the {\it outer degree (or outer curvature)} of the graph is bounded from above, and we allow the potential to go to zero at infinity in a controlled way. Finally, we discuss the optimality of the conditions on the potential and on the outer degree on special graphs.

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arXiv2023

Uniqueness for local-nonlocal elliptic equations

We study mixed local and nonlocal elliptic equation with a variable coefficient $ρ$. Under suitable assumptions on the behaviour at infinity of $ρ$, we obtain uniqueness of solutions belonging to certain weighted Lebsgue spaces, with a weight depending on the coefficient $ρ$. The hypothesis on $ρ$ is optimal; indeed, when it fails we get nonuniqueness of solutions. We also investigate the parabolic counterpart of such equation.

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arXiv2023

Nonexistence results for semilinear elliptic equations on weighted graphs

We study semilinear elliptic inequalities with a potential on infinite graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we prove that the problem does not admit any nonnegative nontrivial solution. We also show that our conditions are optimal.

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arXiv2023

A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is $u\equiv 0$. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

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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

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 $Λ$ of the $L^2$ spectrum of $-Δ$ 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 $Λ$. 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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arXiv2022

Blow-up for semilinear parabolic equations in cones of the hyperbolic space

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{μt}u^p$ ($μ\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $μ$ and $p$, related to the bottom of the spectrum of $-Δ$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $μ$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters $μ$ and $p$ are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting, and it is essentially due to a specific geometric feature of $\mathbb H^n$.

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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

Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains

We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Ω\times(0,+\infty)$, where $Ω$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.

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arXiv2021

Global Solutions of Semilinear Parabolic Equations with Drift Term on Riemannian Manifolds

We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.

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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

Uniqueness for degenerate parabolic equations in weighted $L^1$ spaces

We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an appropriate integral condition; in particular, such condition holds for possibly unbounded solutions belonging to a suitable weighted $L^1$ space.

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arXiv2020

Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that $ρ(x)\sim |x|^{-q}$ as $|x|\to +\infty$ with $q\ge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of {\it slowly} decaying density at infinity, i.e. $q\in [0,2)$, is examined in [41].

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arXiv2020

Blow-up and global existence for the inhomogeneous porous medium equation with reaction

We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term. We show that for small enough initial data, if $ρ(x)\sim \frac{1}{\left(\log|x|\right)^α|x|^{2}}$ as $|x|\to \infty$, then solutions globally exist for any $p>1$. On the other hand, when $ρ(x)\sim\frac{\left(\log|x|\right)^α}{|x|^{2}}$ as $|x|\to \infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $ρ(x)\sim |x|^{-q}$ for $q>0$ as $|x|\to \infty$.

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arXiv2020

Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density

We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $ρ(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $ρ(x)\lesssim |x|^{-q}$ as $|x|\to +\infty$ with $q\in [0, 2).$ We show that for large enough initial data, solutions blow-up in finite time for any $p>1$. On the other hand, if the initial datum is small enough and $p>\bar p$, for a suitable $\bar p$ depending on $ρ, m, N$, then global solutions exist. In addition, if $p<\underline p$, for a suitable $\underline p\leq \bar p$ depending on $ρ, m, N$, then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypotehsis that $q\in [0, ε)$ for $ε>0$ small enough, when $m\leq p<\underline p$. Observe that $\underline p=\bar p$, if $ρ(x)$ is a multiple of $|x|^{-q}$ for $|x|$ large enough. Such results are in agreement with those established in [41], where $ρ(x)\equiv 1$. The case of fast decaying density at infinity, i.e. $q\geq 2$, is examined in [31].

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