arXiv subjects

Juan Luis Vázquez

Publications and source records attributed to Juan Luis Vázquez.

At least 37 records · Page 2Linked to original sources

arXiv2021

The fractional p-Laplacian evolution equation in $\mathbb{R}^N$ in the sublinear case

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1 2$ has been dealt with in a recent paper. We study here the "fast" regime $1<p<2$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value $M$ and any $p$ in the subrange $p_c=2N/(N+s)<p<2$, and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $L^q$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $p_c<p<p_1$, where $p_1$ is a new critical number that we introduce, $p_1\in (p_c,2)$. We extend this type of singular solutions to the "very fast range" $1<p<p_c$. They represent examples of weak solutions having finite-time extinction in that lower $p$ range. We briefly examine the situation in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.

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arXiv2020

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain $Ω$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial Ω$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

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arXiv2020

The evolution fractional p-Laplacian equation in $\mathbb{R}^N$. Fundamental solution and asymptotic behaviour

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interesting properties. As main results, we construct the self-similar fundamental solution for every mass value $M,$ and prove that general finite-mass solutions converge towards that fundamental solution having the same mass in all $L^q$ spaces.A number of additional properties and estimates complete the picture.

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arXiv2020

Estimates on translations and Taylor expansions in fractional Sobolev spaces

In this paper we study how the (normalised) Gagliardo semi-norms $[u]_{W^{s,p} (\mathbb{R}^n)}$ control translations. In particular, we prove that $\| u(\cdot + y) - u \|_{L^p (\mathbb{R}^n)} \le C [ u ] _{W^{s,p} (\mathbb{R}^n)} |y|^s$ for $n\geq1$, $s \in [0,1]$ and $p \in [1,+\infty]$, where $C$ depends only on $n$. We then obtain a corresponding higher-order version of this result: we get fractional rates of the error term in the Taylor expansion. We also present relevant implications of our two results. First, we obtain a direct proof of several compact embedding of $W^{s,p}(\mathbb{R}^n)$ where the Fréchet-Kolmogorov Theorem is applied with known rates. We also derive fractional rates of convergence of the convolution of a function with suitable mollifiers. Thirdly, we obtain fractional rates of convergence of finite-difference discretizations for $W^{s,p} (\mathbb{R}^n))$.

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arXiv2020

Separatrices in the Hamilton-Jacobi Formalism of Inflaton Models

We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lema\^ıtre-Robertson-Walker universe. The existence and properties of separatrices are investigated in the framework of the Hamilton-Jacobi formalism, where the main quantity is the Hubble parameter considered as a function of the inflaton field. A wide class of inflaton models that have separatrix solutions (and include many of the most physically relevant potentials) is introduced, and the properties of the corresponding separatrices are investigated, in particular, asymptotic inflationary stages, leading approximations to the separatrices, and full asymptotic expansions thereof. We also prove an optimal growth criterion for potentials that do not have separatrices.

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arXiv2020

On a fractional thin film equation

This paper deals with a nonlinear degenerate parabolic equation of order $α$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $α=4$ while the Porous Medium Equation is the limit $α=2$. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

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arXiv2020

On the two-phase fractional Stefan problem

The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest for us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

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arXiv2019

Singular boundary behaviour and large solutions for fractional elliptic equations

We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false for equations driven by a wide class of nonlocal fractional operators, extending previous findings for some models of the fractional Laplacian operator. Actually, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal pperators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function.

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arXiv2019

The fractional Schrödinger equation with singular potential and measure data

We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $Ω$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-Δ)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of $(-Δ)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-Δ)^s u + V u = μ$ exists for every Borel measure $μ$. On the other hand, when $V$ is allowed to be more singular, but only on a finite set of points, a solution of $(-Δ)^s u + V u = δ_x$, where $δ_x$ is the Dirac measure at $x$, exists if and only if $h(y) = V(y) |x - y|^{-(n+2s)}$ is integrable on some small ball around $x$. We prove that the set $Z = \{x \in Ω: \textrm{no solution of } (-Δ)^s u + Vu = δ_x \textrm{ exists}\}$ is relevant in the following sense: a solution of $(-Δ)^s u + V u = μ$ exists if and only if $|μ| (Z) = 0$. Furthermore, $Z$ is the set points where the strong maximum principle fails, in the sense that for any bounded $f$ the solution of $(-Δ)^s u + Vu = f$ vanishes on $Z$.

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arXiv2019

Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

For a fixed bounded domain $D \subset \mathbb{R}^N$ we investigate the asymptotic behaviour for large times of solutions to the $p$-Laplacian diffusion equation posed in a tubular domain \begin{equation*} \partial_t u = Δ_p u \quad \text{ in } D \times \mathbb{R}, \quad t > 0 \end{equation*} with $p>2$, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level $u = 0$ and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in $D\subset \mathbb{R}^N$. We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times. The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework $p=2$ solutions behave quite differently.

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arXiv2019

Asymptotic behaviour for the Heat Equation in Hyperbolic Space

Following the classical result of long-time asymptotic convergence towards the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question of long-time behaviour of the Heat Equation in the Hyperbolic Space $\mathbb{H}^n$, $n>1$, also for integrable solutions. We show that the typical convergence proof towards the fundamental solution works in the class of radially symmetric solutions. We also prove the more precise result that says that this limit behaviour is exactly described by the 1D Euclidean kernel, but only after correction of a remarkable outward drift with constant speed produced by the geometry. Finally, we find that such fine convergence results are false for general nonnegative solutions with integrable initial data.

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arXiv2019

Existence of weak solutions for a general porous medium equation with nonlocal pressure

We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-Δ)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0 1$ by developing a new approximation method that allows to treat the range $m\ge 3$ that could not be covered by previous works. We also extend the class of initial data to include any non-negative measure $μ$ with finite mass. In passing from bounded initial data to measure data we make strong use of an $L^1$-$L^\infty$ smoothing effect and other functional estimates. Finite speed of propagation is established for all $m\ge 2$, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for $m<2$.

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arXiv2018

Asymptotic behaviour methods for the Heat Equation. Convergence to the Gaussian

In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we proceed with the proofs of convergence to the Gaussian fundamental solution, a result that holds for all integrable solutions, and represents in the PDE setting the Central Limit Theorem of probability. We present several methods of proof: first, the scaling method. Then several versions of the representation method. This is followed by the functional analysis approach that leads to the famous related equations, Fokker-Planck and Ornstein-Uhlenbeck. The analysis of this connection is also given in rather complete form here. Finally, we present the Boltzmann entropy method, coming from kinetic equations. The different methods are interesting because of the possible extension to prove the asymptotic behaviour or stabilization analysis for more general equations, linear or nonlinear. It all depends a lot on the particular features, and only one or some of the methods work in each case.Other settings of the Heat Equation are briefly discussed in Section 9 and a longer mention of results for different equations is done in Section 10.

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arXiv2018

The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation $(-Δ)^s u + V u = f$ posed in bounded domain $ Ω\subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L^1_{loc}(Ω)$ and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions $f$. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration. Related literature is commented.

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arXiv2018

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the Porous Medium Equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan-Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted Porous Medium Equations in the Euclidean space having special weights.

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arXiv2018

Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications

We consider a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. We establish an equivalence between this equation and the standard $p$-parabolic equation posed in a fictitious space dimension, valid for radially symmetric solutions. This allows us to find suitable explicit solutions for example of Barenblatt type, and as a consequence we settle the exact asymptotic behaviour of the Cauchy problem even for nonradial data. We also establish the asymptotic behaviour in a bounded domain. Moreover, we use the explicit solutions to establish the parabolic Harnack's inequality.

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arXiv2018

Porous medium equation with nonlocal pressure

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-Δ)^{-s}u)$, which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0 2$, and the asymptotic behavior of solutions when $N=1$. The cases $m = 1$ and $m = 2$ were rather well known.

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arXiv2017

Asymptotic behaviour for the Fractional Heat Equation in the Euclidean space

We consider weak solutions of the fractional heat equation posed in the whole $n$-dimensional space, and establish their asymptotic convergence to the fundamental solution as $t\to\infty$ under the assumption that the initial datum is an integrable function, or a finite Radon measure. Convergence with suitable rates is obtained for solutions with a finite first initial moment, while for solutions with compactly supported initial data convergence in relative error holds. The results are applied to the fractional Fokker-Planck equation. Brief mention of other techniques and related equations is made.

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