Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.Comment: The results of this paper were presented by the first author on September 30th, 2015 in the Analysis Seminar of the Department of Mathematics at The University of Texas at Austin. The paper was first submitted to arXiv on November 5th, 2015. We include new estimates and an Almgren monotonicity formula for fractional caloric functions. 27 pages. To appear in SIAM Journal of Mathematical Analysi

A Nondivergence Parabolic Problem with a Fractional Time derivative

We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.Comment: Final version accepted for publication in Differential and Integral Equation

The mathematical theories of diffusion. Nonlinear and fractional diffusion

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 development, the theory of stochastic differential equations gives a foundation to the probabilistic study of diffusion. Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. We use the porous medium and fast diffusion equations as case examples. A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.Comment: To appear in Springer Lecture Notes in Mathematics, C.I.M.E. Subseries, 201

The Calder\'on problem for a space-time fractional parabolic equation

In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation.Comment: 34 page

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 $\partial_t u +(-\Delta)^s u=0$, $0<s<1$, 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 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.Comment: 27 page

Getting acquainted with the fractional Laplacian

These are the handouts of an undergraduate minicourse at the Universit\`a di Bari, in the context of the 2017 INdAM Intensive Period "Contemporary Research in elliptic PDEs and related topics". Without any intention to serve as a throughout epitome to the subject, we hope that these notes can be of some help for a very initial introduction to a fascinating field of classical and modern research.Comment: updated version, 72 pages, 12 figure

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=\Delta u-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0<s<1$.Comment: 23 pages. To appear in Communications on Pure and Applied Analysi

Regularity for parabolic systems with critical growth in the gradient and applications

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove H\"older regularity of weak solutions

On the parabolic Harnack inequality for non-local diffusion equations

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $d\ge\beta$, where $\beta\in(0,2]$ is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $t>0$ in dimensions $d\ge\beta$. This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $u_0 \in L^q_{loc}$ for $q$ larger than the critical value $\tfrac d\beta$ of the elliptic operator $(-\Delta)^{\beta/2}$, a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $d=1$ provided $\beta>1$, since we prove that the local Harnack inequality holds if $d<\beta$.Comment: 21 page

Representation of solutions and large-time behavior for fully nonlocal diffusion equations

We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal $L^2$-decay of mild solutions in all dimensions, (iv) $L^2$-decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii). Next we use Fourier analysis techniques to obtain the optimal decay rate for mild solutions. Here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions. Consequently, the decay rate does not anymore improve when the dimension increases. The theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis. Finally, we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation. Our main idea is to show that the $L^2$-norm is actually a subsolution to a purely time-fractional problem which allows us to use the known theory to obtain the result