Related papers: A fractal local smoothing problem for the wave equ…

We investigate an $L_{q}(L_{p})$-regularity ($1<p,q<\infty$) theory for space-time nonlocal equations of the type $\partial^{\alpha}_{t}u = \mathcal{L}u +f$. Here, $\partial^{\alpha}_{t}$ is the Caputo fractional derivative of order…

We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with Muckenhoupt weights for the equation $$ \partial_{t}^{\alpha}u(t,x)=\Delta u(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $\alpha\in (0,2)$ and…

This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of a suitable…

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,\\…

The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schr\"odinger equation. The fact that the potential term…

This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the…

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition…

The purpose of this article is to review the developments related to the notion of local fractional derivative introduced in 1996. We consider its definition, properties, implications and possible applications. This involves the local…

It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed.

In this paper, we are interested in the $L^p$-estimates of the Boltzmann equation in the case that the distribution function stays around a travelling local Maxwellian. For this, we divide both sides of the Boltzmann equation by the…

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any…

We consider the scattering problem on locally perturbed periodic penetrable dielectric layers, which is formulated in terms of the full vector-valued time-harmonic Maxwell's equations. The right-hand side is not assumed to be periodic. At…

This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $\alpha \in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the…

We prove weighted $L^2$ and refined $L^p$ decoupling estimates for functions whose Fourier transforms are supported in a small neighborhood of the unit sphere or the truncated paraboloid with an additional lower-dimensional frequency…

We consider an evolution equation whose time-diffusion is of fractional type and we provide decay estimates in time for the $L^s$-norm of the solutions in a bounded domain. The spatial operator that we take into account is very general and…

In the realm of complexity, it is argued that adequate modeling of TeV-physics demands an approach based on fractal operators and fractional calculus (FC). Non-local theories and memory effects are connected to complexity and the FC. The…

We develop a wavelet like representation of functions in $L^p(\mathbb{R})$ based on their Fourier--Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the…

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic…

We present a maximal $L_{q}(L_{p})$-regularity theory with Muckenhoupt weights for the equation \begin{equation}\label{eqn 01.26.16:00} \partial^{\alpha}_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in\mathbb{R}^{d}.…

We examine the fractional heat diffusion equations $L_{\gamma,a}:=(-\Delta_a)^{\frac{\gamma}{2}}+\partial_t$, where $\Delta_a$ is the Laplace- or the Bessel-Laplace operator. We give conditions for removability which are sufficient and…