Related papers: Too much regularity may force too much uniqueness

In this work, for an equation of high even order with a fractional derivative in the sense of Caputo, a problem is studied in a rectangular domain with conjugation conditions. A criterion for the uniqueness of a solution is given. The…

We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset $\Omega$ $\subseteq$ R n. We obtain maximal regularity in L 2…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in Caputo's sense,…

Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0,1)$ such that in the limit case ($\alpha =1$) both problems coincide with the same classical Stefan problem. For the one…

For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein--Uhlenbeck equation $$\begin{cases} (-\Delta+x\cdot\nabla)^su=f&\hbox{in}~\Omega\\ u=0&\hbox{on}~\partial\Omega, \end{cases}$$ where $\Omega$ is a possibly…

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative…

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\…

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular…

A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this…

Given a fractional differential equation of order $\alpha \in (0,1]$ with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two…

We study the generalized fractional linear problem $D^{\nu}_{a+*} f(x) =A(x)f(x)+g(x)$, where $D^{\nu}$ is an arbitrary mixture of Caputo derivatives of order at most one and $A(x)$ a family of operators in a Banach space generating…

In this paper we identify, for small $t$ and a fixed $T>0,$ the order $\alpha>0$ in the abstract fractional differential equation $$\partial^\alpha u(t)=Au(t),$$ where the time-fractional derivative $\partial^\alpha$ is understood in the…

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result…

We generalize the fractional Caputo derivative to the fractional derivative ${^CD^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional derivative…

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional…

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…

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation $$ u_{tt}-\mu(t)\Delta u+\omega(t)u_t=f(u),\ x\in\Omega,\ t\in\mathbb{R}, $$ subject to Dirichlet boundary condition…

Let $1< \alpha <2$ and $A$ be the generator of an $\alpha$-times resolvent family $\{S_\alpha(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\bf D}_t^\alpha u(t) = Au(t)+f(t)$, $t \in [0,r]$; $u(0),…

We begin with a brief overview of the most commonly used fractional derivatives, namely the Caputo and Riemann-Liouville derivatives. We then focus on the study of the fractional time wave equation with the Riemann-Liouville derivative,…

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_\Omega F(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or…