Abstract:
The corrosion performance of galvanized steel in closed rusty seawater (CRS) was investigated using weight loss, Tafel polarization curve, and electrochemical impedance spectroscopy. Scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS) were carried out for morphological and chemical characterization of the rust layer absorbed on the zinc coating. Effects of temperature and hydrostatic pressure on corrosion resistance of galvanized steel were studied. Results indicated that rust layer could induce pitting corrosion on the zinc coating under the Cl？ erosion; high temperature accelerated the corrosion rate of zinc coating and inhibited the absorption of rust layer; the polarization resistance ( ) of galvanized steel increased with the increase of hydrostatic pressure in CRS. 1. Introduction The PVC encapsulated galvanized steel wire is widely applied in submarine cable project in recent years [1]. PVC is coated on the surface of galvanized steel by thermal compression. This PVC coating can prevent the penetration of seawater and air and then inhibit the corrosion of galvanized steel. However, the failure of PVC will result in the penetration of seawater with immersion time, and the initial rust layer produced on the surface of the galvanized steel cannot fall off because the coating of PVC. Thus, the galvanized steel was always immersed in a closed saturated rusty seawater environment (Figure 1). Figure 1: Schematic diagram of galvanized steels coated by PVC: (a) the failure zone of the PVC coating, (b) the corroded surface of the galvanized steel. It is acknowledged that when galvanized steel is exposed to marine environment, the main corrosion products (Zn5(OH)2Cl8) have been found in longer exposure periods [2, 3]. When the galvanized steel is coated by PVC and used as cables in seawater, the corrosion performance of galvanized steel under this closed rusty seawater (CRS) is primary in the process of metallic corrosion [4, 5]. It is expected that the corrosion performance of galvanized steel may be different since the rust layer affects corrosion-related processes, such as the mass transport of dissolved oxygen [6, 7], the stability of the passive film, and the hydration of the dissolved metal ions [8–10]. There are few reports regarding the corrosion behavior of galvanized steel under closed rusty seawater environment. In our previous works, it had been demonstrated that Cl？ concentration and pH have very important effects on corrosion behavior of galvanized steel under simulated rust layer solution [11, 12]. In the present work,

Abstract:
The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in these embeddings. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.

Abstract:
In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p\quad &{\rm in}\quad \Omega\setminus\{0\},\\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminus\Omega, \end{array} \end{equation} where $p>1$, $\Omega$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing the origin, $N\ge2$ and the fractional Laplacian $(-\Delta)^\alpha$ is defined in the principle value sense. We obtain that any classical solution $u$ of (\ref{eq 0.1}) is a weak solution of \begin{equation}\label{eq 0.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p+k\delta_0\quad &{\rm in}\quad \Omega,\\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminus\Omega \end{array} \end{equation} for some $k\ge0$, where $\delta_0$ is the Dirac mass at the origin. In particular, when $p\ge \frac{N}{N-2\alpha}$, we have that $k=0$; when $p< \frac{N}{N-2\alpha}$, $u$ has removable singularity at the origin if $k=0$ and if $k>0$, $u$ satisfies $$\lim_{x\to0} u(x)|x|^{N-2\alpha}=c_{N,\alpha}k,$$ where $c_{N,\alpha}>0$. Furthermore, when $p\in(1, \frac{N}{N-2\alpha})$, we obtain that there exists $k^*>0$ such that problem (\ref{eq 0.1}) has at least two positive solutions for $kk^*$.

Abstract:
In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad \Omega,\\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=g\quad & {\rm on}\quad \partial \Omega, \end{array} \end{equation} where $\Omega$ is a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$, $\alpha\in(\frac12,1)$ and the operator $(-\Delta)^\alpha_\Omega$ denotes the regional fractional Laplacian. We prove that when $g\equiv0$, problem (\ref{0}) admits a unique weak solution in the cases that $f\in L^2(\Omega)$, $f\in L^1(\Omega, \rho^\beta dx)$ and $f\in \mathcal{M}(\Omega,\rho^\beta)$, here $\rho(x)={\rm dist}(x,\partial\Omega)$, $\beta=2\alpha-1$ and $\mathcal{M}(\Omega,\rho^\beta)$ is a space of all Radon measures $\nu$ satisfying $\int_\Omega \rho^\beta d|\nu|<+\infty.$ Finally, we provide an Integral by Parts Formula for the classical solution of (\ref{0}) with general boundary data $g$.

Abstract:
The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm] u=0,\quad & \rm{in}\quad \R^N\setminus\Omega,\\[2mm] \lim_{x\to 0}u(x)=+\infty, \end{array} \right. $$ where $p>0$, $\Omega$ is an open, bounded and smooth domain of $\R^N\ (N\ge2)$ with $0\in\Omega$. We analyze the existence, nonexistence, uniqueness and asymptotic behavior of the solutions.

Abstract:
Our purpose of this paper is to classify isolated singularities of positive solutions to Choquard equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle \ \ -\Delta u+ u =I_\alpha[u^p] u^q\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\},\\[2mm] \phantom{ } \qquad \ \displaystyle \lim_{|x|\to+\infty}u(x)=0, \end{array} \end{equation} where $p>0,\, q\ge 1$, $N\ge3$, $\alpha\in(0,N)$ and $$I_\alpha[u^p](x)=\int_{\mathbb{R}^N}\frac{u(y)^p}{|x-y|^{N-\alpha}}\, dy.$$ We show that any positive solution $u$ of (\ref{eq 0.1}) is a weak solution of \begin{equation}\label{eq 0.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle\ \ -\Delta u+u=I_\alpha[u^p]u^q+k\delta_0\quad &{\rm in}\quad \mathbb{R}^N,\\[2mm] \phantom{ } \qquad \ \displaystyle \lim_{|x|\to+\infty}u(x)=0 \end{array} \end{equation} for some $k\ge 0$, where $\delta_0$ is the Dirac mass at the origin. In the supercritical case: $ p+q\ge \frac{N+\alpha}{N-2}$ or $p\ge \frac{N}{N-2}$ or $q\ge \frac{N}{N-2},$ we have that $k=0$. In the subcritical case: $p+q< \frac{N+\alpha}{N-2}$ and $p< \frac{N}{N-2}$ and $q<\frac{N}{N-2}$, $u$ has removable singularity at the origin if $k=0$; if $k>0$, then $\lim_{|x|\to0^+} u(x)|x|^{N-2}=c_{N} k.$ Furthermore, we prove the existence of singular solutions of (\ref{eq 0.1}) in the subcritical case by searching the weak solutions of (\ref{eq 0.2}).

Abstract:
In this paper, we study the existence of nonnegative weak solutions to (E) $ (-\Delta)^\alpha u+h(u)=\nu $ in a general regular domain $\Omega$, which vanish in $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a nonnegative Radon measure and $h:\mathbb{R}_+\to\mathbb{R}_+$ is a continuous nondecreasing function satisfying a subcritical integrability condition. Furthermore, we analyze properties of weak solution $u_k$ to $(E)$ with $\Omega=\mathbb{R}^N$, $\nu=k\delta_0$ and $h(s)=s^p$, where $k>0$, $p\in(0,\frac{N}{N-2\alpha})$ and $\delta_0$ denotes Dirac mass at the origin. Finally, we show for $p\in(0,1+\frac{2\alpha}{N}]$ that $u_k\to\infty$ in $\mathbb{R}^N$ as $k\to\infty$, and for $p\in(1+\frac{2\alpha}{N},\frac{N}{N-2\alpha})$ that $\lim_{k\to\infty}u_k(x)=c|x|^{-\frac{2\alpha}{p-1}}$ with $c>0$, which is a classical solution of $ (-\Delta)^\alpha u+u^p=0$ in $\mathbb{R}^N\setminus\{0\}$.

Abstract:
We study the existence of weak solutions of (E) $ (-\Delta)^\alpha u+g(u)=\nu $ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a Radon measure and $g$ is a nondecreasing function satisfying some extra hypothesis. When $g$ satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where $\nu$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r|^{k-1}r$ with $k$ supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

Abstract:
Let $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$ and $\Omega\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $\delta_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $ (-\Delta)^\alpha u+u^p=k\delta_0 $ in $\Omega$ which vanishes in $\Omega^c$, is a classical solution of $(E_*)$ $ (-\Delta)^\alpha u+u^p=0 $ in $\Omega\setminus\{0\}$ with the same outer data. When $\frac{2\alpha}{N-2\alpha}\leq 1+\frac{2\alpha}{N}$, $p\in(0, 1+\frac{2\alpha}{N}]$ we show that the $u_k$ converges to $\infty$ in whole $\Omega$ when $k\to\infty$, while, for $p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$, the limit of the $u_k$ is a strongly singular solution of $(E_*)$. The same result holds in the case $1+\frac{2\alpha}{N}<\frac{2\alpha}{N-2\alpha}$ excepted if $\frac{2\alpha}{N}

Abstract:
We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where $\epsilon=1$ or $-1$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(1/2,1)$, $\nu$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $\nu$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $\epsilon=1$.