相关论文: Maximal regularity for time-fractional Schr\"oding…

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit…

We consider the Dirichlet problem $\lambda U - {\mathcal{L}}U= F$ in \mathcal{O}, U=0 on $\partial \mathcal{O}$. Here $F\in L^2(\mathcal{O}, \mu)$ where $\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$,…

Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag--Leffler Euler integrator of a class of stochastic…

In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim \chi_{\{u > 0 \}}$. The…

In this work, we are devoted to the reconstruction of an unknown initial value from the terminal data. The asymptotic and root-distribution properties of Mittag-Leffler functions are used to establish stability of the backward problem.…

We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger equation with a general nonlinearity. We prove the local well-posedness of it in fractional Sobolev spaces based on the decay estimates and H\"older type…

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs…

In this paper, we study the almost everywhere convergence of the cubic nonlinear Schr\"odinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^2. \end{equation*} Inspired by…

This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $\Omega\times I $, where $P$ is a nonlocal operator, and $\Omega \subset R^n$,…

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $\Omega$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$,…

In this paper, we study the focusing nonlinear Schr\"odinger equation with exponential nonlinearities \[ i \partial_t u + \Delta u = - \left(e^{4\pi |u|^2} - 1 - 4\pi \mu |u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}…

Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive analytic semigroups on UMD-Banach lattices, namely $\ell_p(\ell_q)$ for $p \neq q \in (1, \infty)$,…

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the…

This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate…

We consider the Cauchy problem for the $L^2$-critical nonlinear Schr\"odinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup…

Using Guth's polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schr\"odinger equation in $\mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result…

The spectrum of a Schr\"odinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of…