arXiv subjects

Matteo Bonforte

Publications and source records attributed to Matteo Bonforte.

At least 37 records · Page 2Linked to original sources

arXiv2019

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=Δu^m$, posed in a smooth bounded domain $Ω\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2) 0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincaré inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of "almost orthogonality", which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows.

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arXiv2018

Quantitative a Priori Estimates for Fast Diffusion Equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^γ \nabla\cdot (|x|^{-β} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb R}^N$. The weights $|x|^γ$ and $|x|^{-β}$, with $γ< N$ and $γ-2 < β\leq γ(N-2)/N$ can be both degenerate and singular and need not belong to the class $\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Hölder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.

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arXiv2018

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $Ω\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-Δ)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a power function $f(u)\sim u^p$, with $p\le 1$. The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number $\frac{2s}{1-p}$ goes below the exponent $γ\in(0,1]$ corresponding to the Hölder regularity of the first eigenfunction $\mathcal{L}Φ_1=λ_1 Φ_1$. Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless γ$, and in particular a logarithmic correction appears in the "critical" case $\frac{2s}{1-p} = γ$. Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless γ$, and in particular a logarithmic correction appears in the "critical" case $\frac{2s}{1-p} = γ$. For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $0<s\leq \frac{1-p}{2}$.

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arXiv2017

Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains

This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω\subset{\mathbb R}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, ${\mathcal L}$ can be a power of a uniformly elliptic operator with $C^1$ coefficients. Since the nonlinearity is given by $u^m$ with $m>1$, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations has been recently developed in [14,15]. Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when ${\mathcal L}$ is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of non-matching powers for the long-time boundary behavior. More precisely, when ${\mathcal L}=(-Δ)^s$ is a spectral power of the {Dirichlet} Laplacian inside a smooth domain, we can prove that: - when $2s> 1-1/m$, for large times all solutions behave as ${\rm dist}^{1/m}$ near the boundary; - when $2s\le 1-1/m$, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation ${\mathcal L}u^m=u$.

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arXiv2016

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-Δ)^s u=0$, $0<s<1$, posed in the whole space $\mathbb{R}^N$ with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in particular we prove optimal pointwise estimates and new Harnack inequalities.

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arXiv2016

Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Δ)^su^m=0$ in $(0,\infty)\timesΩ$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminusΩ)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partialΩ$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,α}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $Ω$, both in bounded or unbounded domains.

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arXiv2016

Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized.

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arXiv2016

Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{é} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

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arXiv2015

Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\inΩ\subset \mathbb{R}^N$, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-Δ)^s$, $0 1$. In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-$L^1$ estimates. The class of solutions is very well suited for that work. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit. In a companion paper [12], we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed.

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arXiv2015

Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-Δ)^s ϕ(u)= 0,$ with $0 0$, or $ϕ(u) = \log u$, and we take nonnegative $L^1$ initial data, there is no (nonnegative) solution of the problem in any dimension $N\ge 2$. We find the range of non-existence when $N=1$ in terms of $s$ and $n$. The range of exponents that we find for non-existence both for parabolic and elliptic equations are optimal. Non-existence is then proved for more general nonlinearities $ϕ$, and it is also extended to the related elliptic problem of nonlinear nonlocal type: $u + (-Δ)^s ϕ(u) = f$ with the same type of nonlinearity $ϕ$.

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arXiv2014

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

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arXiv2013

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains

We investigate quantitative properties of the nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + {\mathcal L} (u^m)=0$, posed in a bounded domain, $x\inΩ\subset {\mathbb R}^N$ with $m>1$ for $t>0$. As ${\mathcal L}$ we use one of the most common definitions of the fractional Laplacian $(-Δ)^s$, $0<s<1$, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. The standard Laplacian case $s=1$ or the linear case $m=1$ are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.

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arXiv2013

Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations

We establish quantitative estimates for solutions $u(t,x)$ to the fractional nonlinear diffusion equation, $\partial_t u +(-Δ)^s (u^m)=0$ in the whole range of exponents $m>0$, $0<s<1$. The equation is posed in the whole space $x\in\mathbb{R}^d$. We first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data. In the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions, depending only on the norm of the initial data in a certain ball. The estimates take a different form in three exponent ranges: slow diffusion, good range of fast diffusion, and very fast diffusion. Finally, we show existence and uniqueness of initial traces.

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arXiv2012

Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations

The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form $-Δu= c u^p$, with $0<p<p_s=(d+2)/(d-2)$, defined on bounded domains of $\RR^d$, $d\ge 3$, without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constant, as well as gradient bounds.

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arXiv2011

Total Variation Flow and Sign Fast Diffusion in one dimension

We consider the dynamics of the Total Variation Flow (TVF) $u_t=÷(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=Δ\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.

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arXiv2011

Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space

We study the Emden-Fowler equation $-Δu=|u|^{p-1}u$ on the hyperbolic space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is $p=(n+2)/(n-2)$ as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers \cite{mancini, bhakta} consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.

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arXiv2010

Behaviour near extinction for the Fast Diffusion Equation on bounded domains

We consider the Fast Diffusion Equation $u_t=Δu^m$ posed in a bounded smooth domain $Ω\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive solutions $u(t,x)$ of such problem extinguish in a finite time $T$, and also that such solutions approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. Here we are interested in describing the behaviour of the solutions near the extinction time. We first show that the convergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow. For $m$ close to 1 we get such rates by means of entropy methods and weighted Poincaré inequalities. The analysis of the latter point makes an essential use of fine properties of the associated stationary elliptic problem $-ΔS^m= {\bf c} S$ in the limit $m\to 1$, and such a study has an independent interest.

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arXiv2009

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.

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