On the radius of spatial analyticity for the modified Kawahara equation on the line

First, by using linear and trilinear estimates in Bourgain type analytic and Gevrey spaces, the local well‐posedness of the Cauchy problem for the modified Kawahara equation on the line is established for analytic initial data 40-91db-47ae-a50c-c482b0faab71/mana201800394-math-0001.png"> that can be extended as holomorphic functions in a strip around the x‐axis. Next we use this local result and a Gevrey approximate conservation law to prove that global solutions exist. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity given by , where can be taken arbitrarily small and c is a positive constant.     

Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators

We consider a Schrödinger operator $L=?d^2/d{x}^2+V(x)$ on $\mathbb{R}$, where V is a real-valued measurable function, and give an explicit and simple characterization of intrinsic ultracontractivity (IU) of the Schrödinger semigroup generated by L for a wide class of potentials. By making use of it, we also give new examples of potentials for which the semigroups satisfy (IU) or non-(IU).     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Preparation and characterization of new inorganic–organic hybrid catalyst H3PMo12O40/Hyd‐SBA‐15 and its application in the domino multi‐component reaction

We present novel inorganic–organic hybrid catalyst to accomplish domino multi‐component reaction (MCR) for synthesis of 3‐amino‐2′‐oxospiro[benzo[c]pyrano[3,2‐a]phenazine‐1,3′‐indoline]‐2‐carbonitrile/carboxylate derivatives. This methodology offers remarkable development by easy production of H₃PMo₁₂O₄₀/Hyd‐SBA‐15 in regard to solving the problem of using harsh catalysts, also it demonstrates to be impressive and environmentally friendly in term of low reaction times and high yields.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

The Lp dual Minkowski problem for p > 1 and q > 0

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. $L_p$ dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_p$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.     

Travelling waves in dilatant non-Newtonian thin films

We prove the existence of a travelling wave solution for a gravity-driven thin film of a viscous and incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second order partial differential equations for the film's height h and the surfactant's concentration γ are derived by means of lubrication theory applied to the non-Newtonian Navier–Stokes equations.     

Asymptotic behavior of solutions to the compressible nematic liquid crystal system in ℝ3

In this paper, we study a nematic liquid crystals system in three-dimensional whole space ?³ and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L¹?³ additionally.