# Juan Luis VazquezUniversidad Autónoma de Madrid | UAM · Department of Mathematics

Juan Luis Vazquez

Ph.D. in Mathematics, 1979

## About

420

Publications

100,657

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

16,340

Citations

Introduction

Juan Luis Vázquez is Professor of Applied Mathematics at Universidad Autonoma de Madrid, Spain, since 1986. Interested in Nonlinear Elliptic and Parabolic PDES, Free Boundaries, Selfsimilarity, Asymptotic Behaviour. Recent emphasis on fast diffusion, entropy functionals and fractional diffusion operators. Author of the book "The Porous Medium Equation" among others.
Research ties mainly with US, France and Italy.
Fellow of the AMS and EURASC.

Additional affiliations

January 1981 - present

Education

September 1976 - February 1979

## Publications

Publications (420)

The standard problem for the classical heat equation posed in a bounded domain Ω of Rn 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 soluti...

We consider an aggregation-diffusion model, where the diffusion is nonlinear of porous medium type and the aggregation is governed by the Riesz potential of order s. The addition of a quadratic diffusion term produces a more precise competition with the aggregation term for small s, as they have the same scaling if s=0. We prove existence and uniqu...

We consider density solutions for gradient flow equations of the form u t = ∇ · ( γ ( u )∇ N( u )), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ ( u ) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leadin...

We establish existence, uniqueness as well as quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+Ls,p(u)=0, where Ls,p=(−Δ)ps is the standard fractional p-Laplacian operator. We work in the range of exponents 0<s<1 and 1<p<2, and in some sections we need sp<1. The equation is posed in the whole space x∈R...

Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effec...

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova...

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter \(1<p<2\) and fractional exponent \(s\in (0,1)\). Rather standard theory shows that the Cauchy Problem for data in the Lebesgue \(L^q\) spaces is well posed, and the solutions form a family of non-expansive semigroups with regul...

We introduce three representation formulas for the fractional p -Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinea...

The article is dedicated to recalling the life and mathematics of Louis Nirenberg, a distinguished Canadian mathematician who recently died in New York, where he lived. An emblematic figure of Analysis and Partial Differential Equations in the last century, he was awarded the Abel Prize in 2015. From his watchtower at the Courant Institute in New Y...

We study an anisotropic, possibly non-homogeneous version of the evolution 𝑝-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive sharp L 1 L^{1} - L ∞ L^{\infty} estimates. We prove the existence of a self-similar fundamental solution of this equation in t...

We study an anisotropic, possibly non-homogeneous version of the evolution $p$-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive $L^1$ to $L^\infty$ estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropri...

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) and their embeddings, for $$s \in (0,1]$$ s ∈ ( 0 , 1 ] and $$p\ge 1$$ p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodecki...

Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effec...

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional $p$-Laplacian operator. We work in the range of exponents $0<s<1$ and $1<p<2$, and in some sections $sp<1$. T...

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] c...

In this paper we study how the (normalised) Gagliardo semi-norms [u]Ws,p(Rn) control translations. In particular, we prove that ‖u(⋅+y)−u‖Lp(Rn)≤C[u]Ws,p(Rn)|y|s for n≥1, s∈[0,1] and p∈[1,+∞], 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 expans...

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu=λu+f involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green function and the standard eigenbasis of the operator. The main e...

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularit...

We introduce three representation formulas for the fractional $p$-Laplace operator in the whole range of parameters $0<s<1$ and $1<p<\infty$. Note that for $p\ne 2$ this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the non...

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameters p>2 and s∈(0,1) (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue Lq spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interestin...

This note is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have been lately considered by Constantin and Ignatova in the framework of the SQG equation \cite{CI} in bounded domains an...

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ 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...

Our aim is to characterize the homogeneous fractional Sobolev-Slobodecki\u{\i} spaces $\mathcal{D}^{s,p} (\mathbb{R}^n)$ and their embeddings, for $s \in (0,1]$ and $p\ge 1$. They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo-Slobodecki\u{\i} seminorms. For $s\,p < n$ or $s =...

We consider density solutions for gradient flow equations of the form $u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm N(u))$, where $\mathrm N$ is the Newtonian repulsive potential in the whole space $\mathbb R^d$ with the nonlinear convex mobility $\gamma(u)=u^\alpha$, and $\alpha>1$. We show that solutions corresponding to compactly supported init...

We construct the self-similar fundamental solution of the anisotropic porous medium equation in the suitable fast diffusion range. We also prove the asymptotic behavior of finite mass solutions. Positivity, decay rates, as well as other properties are derived.

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 correspondin...

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 othe...

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems $\mathcal L u = \lambda u + f$ involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green's function and the standard eigenbasis of...

We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann–Lemaître–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...

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...

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 to us. Then we introduce the fractional Ste...

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 St...

For a fixed bounded domain D⊂RN we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domain∂tu=Δpu in D×R,t>0 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 exist...

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. The equation for the enthalpy $h$ reads $\partial_t h+ (-\Delta)^s \Phi(h) =0$ where the temperature $u:=\Phi(h):=\max\{h-L,0\}$ is defined for some constant $L>0$ called the latent heat, and $(-\...

We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the mobility. For linear mobility, $\gamma(u)=u$, the equation and some variations have been proposed as a model for supercondu...

We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lema\^{\i}tre-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 consider...

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 Dirich...

We consider the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$, which describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we assume that the solutions are non-negative and the problem is posed in the whole space. The model has bee...

This paper deals with a nonlinear degenerate parabolic equation of order $\alpha$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $\alpha=4$ while the Porous Medium Equation is the limit $\alpha=2$. We prove existence of a nonnegative weak solution for a general clas...

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 beha...

For a fixed bounded domain D ⊂ R N we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domain ∂ t u = ∆ p u in D × R, t > 0 with p > 2, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables,...

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 = \Delta_p u \quad \text{ in } D \times \mathbb{R}, \quad t > 0 \end{equation*} with $p>2$, i.e., the slow diffusion case, and homog...

We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformula...

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...

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation ut = ∇⋅ (um−1∇(−Δ)−su), which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m > 1 and 0 < s < 1, we assume that the solutions are non-negative, and the problem is posed in the whole sp...

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L^1_{loc}(\Omega)$ and prove well-posedness of very weak solutions when the data are chosen in an...

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 $\Omega\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that in...

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 symmetri...

One of the major problems in the theory of the porous medium equation is the regularity of the solutions and the free boundaries. Here we assume flatness of the solution in space time cylinder and derive smoothness of the interface after a small time, as well as smoothness of the solution in the positivity set and up to the free boundary for some t...

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel devel...

We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space domain. We prove existence of weak solutions and suitable a priori bounds and regularity estimates.

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...

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 integ...

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel devel...

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on...

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the ma...

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems. The aut...

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 $\Omega\subset \mathbb{R}^N$. As operators $\ma...

We consider nonlinear parabolic equations involving fractional diffusion of
the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an
open problem concerning the existence of solutions for very singular
nonlinearities $\Phi$ in power form, precisely $\Phi'(u)=c\,u^{-(n+1)}$ for
some $0< n<1$. We also include the logarithmic diffu...

The famous Fisher-KPP reaction diffusion model combines linear diffusion with the typical Fisher-KPP reaction term, and appears in a number of relevant applications. It is remarkable as a mathematical model since, in the case of linear diffusion, it possesses a family of travelling waves that describe the asymptotic behaviour of a wide class soluti...

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-\Delta)^s u=0$, $0<s<1$, 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...

The object of study is the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is ut = ∇ (u∇(-Δ)-1/2u). For definiteness, the problem is posed in (x ε RN, t ε R) with nonnegative initial data u(x, 0) that is integrable and decays at infinity. Previous papers hav...

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$
for the nonlinear parabolic equation involving fractional diffusion $\partial_t
u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities
$\phi$ . More precisely, we prove that when $\phi(u)=-1/u^n$ with $n>0$, or
$\phi(u) = \log u$, and we take nonnegative...

The famous Fisher-KPP reaction-diffusion model combines linear diffusion with
the typical KPP reaction term, and appears in a number of relevant applications
in biology and chemistry. It is remarkable as a mathematical model since it
possesses a family of travelling waves that describe the asymptotic behaviour
of a large class solutions $0\le u(x,t...

We study a porous medium equation with fractional potential pressure:
$$
\partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-\Delta)^{-s}u,
$$
for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. The problem is posed for $x\in \mathbb{R}^N$, $N\geq 1$, and $t>0$. The initial data $u(x,0)$ is assumed to be a bounded function with compact support or fast decay...

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, $x\in\Omega\subset \mathbb{R}^N$,
with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we
can use a quite general class of linear operator...

This is an introduction to The Theme Issue on "Free Boundary Problems and
Related Topics", which consists of 14 survey/review articles on the topics, of
Philosophical Transactions of the Royal Society A: Physical, Mathematical and
Engineering Sciences, 373, no. 2050, The Royal Society, 2015.

We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Lap...

We consider a model of fractional diffusion involving the natural nonlocal
version of the $p$-Laplacian operator. We study the Dirichlet problem posed in
a bounded domain $\Omega$ of ${\mathbb{R}}^N$ with zero data outside of
$\Omega$, for which the existence and uniqueness of strong nonnegative
solutions is proved, and a number of quantitative pro...

We report on recent progress in the study of evolution processes involving
degenerate parabolic equations what may exhibit free boundaries. The equations
we have selected follow to recent trends in diffusion theory: considering
anomalous diffusion with long-range effects, which leads to fractional
operators or other operators involving kernels with...

We study the large-time behaviour of the
nonlinear oscillator
\[
\hskip-20mm m\,x'' + f(x') + k\,x=0\,,
\]
where m, k>0 and f is a monotone real function representing
nonlinear friction. We are interested in understanding the
long-time effect of a nonlinear damping term, with special
attention to the model case $f(x')= A\,|x'|^{\alpha-1}x'$ with
α...

We construct the fundamental solution of the Porous Medium Equation posed in
the hyperbolic space $H^n$ and describe its asymptotic behaviour as
$t\to\infty$. We also show that it describes the long time behaviour of
integrable nonnegative solutions, and very accurately if the solutions are also
radial and compactly supported. By radial we mean fun...

We analyse the asymptotic behaviour of solutions to the one dimensional
fractional version of the porous medium equation introduced by Caffarelli and
V\'azquez, where the pressure is obtained as a Riesz potential associated to
the density. We take advantage of the displacement convexity of the Riesz
potential in one dimension to show a functional i...

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-ti...

We investigate the behaviour of the solutions $u_m(x,t)$ of the fractional
porous medium equation $$ u_t+(-\Delta)^s (u^m)=0, \quad x\in {\mathbb{R}}^N, \
t>0. $$ with initial data $u(x,0)\ge 0$, $x\in {\mathbb{R}}^N$, in the limit as
$m\to\infty$ with fixed $s\in (0,1)$. We first identify the limit of the
Barenblatt solutions as the solution of a...

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a “mean field” equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limi...

We consider four different models of nonlinear diffusion equations involving
fractional Laplacians and study the existence and properties of classes of
self-similar solutions. Such solutions are an important tool in developing the
general theory. We introduce a number of transformations that allow us to map
complete classes of solutions of one equa...

We report on recent progress in the study of nonlinear diffusion equations
involving nonlocal, long-range diffusion effects. Our main concern is the
so-called fractional porous medium equation, $\partial_t u
+(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual
porous medium flows, the fractional version has infinite speed of p...

We report on recent progress in the study of nonlinear diffusion equations in which the author has been involved. The main topic we discuss here is the use of entropy methods to obtain a precise description of the asymptotic behaviour of the solutions of evolution problems posed in the whole space. A detailed account is given of the analysis of the...

We study the regularity properties of the solutions to the nonlinear equation
with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$
posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the
nonlinearity satisfies some not very restrictive conditions: $\varphi\in
C^{1,\gamma}(\mathbb{R})$, $1+\gamma>\sigma$, an...

We study a porous medium equation with fractional potential pressure: $$
\partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-\Delta)^{-s}u, $$ for
$m>1$, $0<s<1$ and $u(x,t)\ge 0$. The problem is posed for $x\in \mathbb{R}^N$,
$N\geq 1$, and $t>0$. The initial data $u(x,0)$ is assumed to be a bounded
function with compact support or fast decay...

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, $x\in\Omega\subset {\mathbb
R}^N$ with $m>1$ for $t>0$. As ${\mathcal L}$ we use one of the most common
definitions of the fractional Laplacian $(-\Delta)...