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In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations where 
\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}
$\nabla\times$ denotes the usual curl operator in 
$\mathbb{R}^3$, 
$\mu_1,\mu_2 \gt 0$, and 
$\beta\in\mathbb{R}\backslash\{0\}$. We show that this critical system admits a non-trivial ground state solution when the parameter β is positive and small. For general 
$\beta\in\mathbb{R}\backslash\{0\}$, we prove that this system admits a non-trivial cylindrically symmetric solution with the least positive energy. We also study the existence of the curl-free solution and the synchronized solution due to the special structure of this system. These seem to be the first results on the critically coupled system containing the curl-curl operator.
In this paper, we study the following high-order elliptic equation involving the GJMS operator:
\begin{align*}\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}.\end{align*}
We establish that if α > 1 and 
$n\geq3$ or if 
$\alpha\in (1-\epsilon_0, 1)$ with 
$n=2m\geq4$, then 
$v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.
We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution u to the scalar wave equation with sufficiently small 
$C_c^\infty $ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for 
$|x|<t$). It involves the scattering data obtained in the author’s asymptotic completeness result in [75]. Using this asymptotic formula, we prove that u must vanish under some decaying assumptions on u or its scattering data, provided that the wave equation violates the null condition.
In this article, we investigate a free boundary problem for the Lotka–Volterra model consisting of an invasive species with density u and a native species with density v in one dimension. We assume that v undergoes diffusion and growth in 
$[0,+\infty )$, and u invades into the environment with spreading front 
$x=h(t)$ satisfying free boundary condition 
$h'(t)=-u_x(t,h(t))-\alpha $ for some decay rate 
$\alpha>0$, this is caused by the bad environment at the boundary. When u is an inferior competitor, 
$u(t,x)$ and 
$h(t)$ tend to 0 within a finite time, while another specie 
$v(t,x)$ tends to a stationary 
$\Lambda (x)$ defined on the half-line. When u is a superior competitor, we have a trichotomy result: spreading of u and vanishing of v (i.e., as 
$t \to +\infty $, 
$h(t)$ goes to 
$+\infty $ and 
$(u,v)\to (\Lambda ,0)$); the transition case (i.e., as 
$t \to +\infty $, 
$(u,v)\to (w_\alpha , \eta _\alpha )$, 
$h(t)$ tends to a finite point); vanishing of u and spreading of v (i.e., 
$u(t,x)$ and 
$h(t)$ tends to 0 within a finite time, 
$v(t,x)$ converges to 
$\Lambda (x)$). Additionally, we show that this trichotomy result depends on the initial data 
$u(0,x)$.
This article is concerned with the following chemotaxis-growth system
\begin{equation*}\begin{cases}u_t = \nabla\cdot \left(\nabla u - u\bf{w} \right) + \rho u - \mu u^\alpha ,&x \in \Omega ,\ t \gt 0,\\v_{1,t} = \Delta v_1 - v_1 + u,&x \in \Omega ,\ t \gt 0,\\v_{2,t} = \Delta v_2 - v_2 + v_1,&x \in \Omega ,\ t \gt 0,\\\ldots \\v_{k,t} = \Delta v_k - v_k + v_{k - 1},&x \in \Omega ,\ t \gt 0,\\{\bf{w}}_t = \Delta {\bf{w}} - {\bf{w}} + \nabla v_k,&x \in \Omega ,\ t \gt 0,\end{cases}\end{equation*}
under the homogeneous Neumann boundary condition for u, vi and the homogeneous Dirichlet boundary condition for 
$\bf{w}$ in a smooth bounded domain 
$\Omega \subset {\mathbb{R}^n}\left( {n \geqslant 1} \right),$ where ρ > 0, µ > 0, α > 1 and 
$i=1,\ldots,k$. We reveal that when the index α, the spatial variable n, and the number of equations k satisfy certain relationships, the global solution of the system exists and converges to the constant equilibrium state in the form of exponential convergence.
In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension,where 
\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}
$\phi *u$ is a spatial convolution with the top hat kernel, 
$\phi (y) \equiv H\left (\frac {1}{4}-y^2\right )$, except that now we modify this to an associated initial-boundary value problem on the finite spatial interval 
$[0,a]$ rather than the whole real line. Boundary conditions are required at the end points of the interval, and we address the situations when these are of either Dirichlet or Neumann type. This model is a natural extension of the classical Fisher-KPP model, with the introduction of the simplest possible nonlocal effect into the saturation term. Nonlocal reaction-diffusion models arise naturally in a variety of (frequently biological or ecological) contexts, and as such it is of fundamental interest to examine their properties in detail, and to compare and contrast these with the well known properties of the classical Fisher-KPP model.
In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire 
$\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.
$\mathbb{R}^3$
We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in 
$\mathbb{R}^3$:which models the motion of swimming bacteria in water flows. First, we prove blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin-type criterion for 
\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}
$\alpha \gt \frac {3}{4}$ and the Beir
$\tilde {\textrm {a}}$o da Veiga-type criterion for 
$\alpha \gt \frac {1}{2}$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for 
$\alpha \geq \frac {5}{4}$. Furthermore, in the scenario of 
$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if only the 
$L^2$-norm of initial data is small. To our knowledge, this work provides the first result concerning the global existence and large-time behaviour of strong solutions for the chemotaxis-Navier–Stokes equations with possibly large oscillations.
In this article, we study the effect of the Hardy potential on existence, uniqueness, and optimal summability of solutions of the mixed local–nonlocal elliptic problem
where Ω is a bounded domain in 
\begin{equation*}-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \,\text{in } \Omega, \ u=0 \,\text{in } {\mathbb R}^n \setminus \Omega,\end{equation*}
${\mathbb R}^n$ containing the origin and γ > 0. In particular, we will discuss the existence, non-existence, and uniqueness of solutions in terms of the summability of f and of the value of the parameter γ.
