In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains.

We consider dual Toeplitz operators on the orthogonal complements of the ock-Sobolev spaces of all nonnegative real orders. First, for symbols in a certain class containing all bounded functions, we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero. Next, for bounded symbols, we construct a symbol map and exhibit a short exact sequence associated with the $C^*$-algebra generated by all dual Toeplitz operators with bounded symbols.

Let $X$ and $Y$ be two normed spaces. Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$. Let $g: X\rightarrow Y$ be a standard $\varepsilon$-phase isometry for some $\varepsilon\geq 0$, i.e., $g(0)=0$, and for all $u,v\in X$,
$$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$$
The mapping $g$ is said to be a phase isometry provided that $\varepsilon=0$.
In this paper, we show the following universal inequality of $g$: for each $u^*\in w^*$-exp $\|u^*\|B_{X^*}$, there exist a phase function $\sigma_{u^*}: X\rightarrow \{-1,1\}$ and $\varphi$ $\in$ $Y^*$ with $\|\varphi\|= \|u^*\|\equiv \alpha $ satisfying that
$$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$$ In particular, let $X$ be a smooth Banach space. Then we show the following:
(1) the universal inequality holds for all $u^*\in X^*$;
(2) the constant $\frac{5}{2}$ can be reduced to $\frac{3}{2}$ provided that $Y^\ast$ is strictly convex;
(3) the existence of such a $g$ implies the existence of a phase isometry $\Theta:X\rightarrow Y$ such that $\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that $Y^{**}$ has the $w^*$-Kadec-Klee property (for example, $Y$ is both reflexive and locally uniformly convex).

By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively.

In this paper, we establish some regularity conditions on the density and velocity fields to guarantee the energy conservation of the weak solutions for the three-dimensional compressible nematic liquid crystal flow in the periodic domain.

We investigate the global existence of strong solutions to a non-isothermal ideal gas model derived from an energy variational approach. We first show the global well-posedness in the Sobolev space $H^{2}\left(\mathbb{R}^{3}\right)$ for solutions near equilibrium through iterated energy-type bounds and a continuity argument. We then prove the global well-posedness in the critical Besov space $\dot{B}_{2,1}^{3 / 2}$ by showing that the linearized operator is a contraction mapping under the right circumstances.

We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale (for short, TCS) model coupled with Stokes equations in the whole space. The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force. In this paper, we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.

This paper is concerned with the following attraction-repulsion chemotaxis system with $p$-Laplacian diffusion and logistic source:
$$\left\{\begin{array}{ll}u_{t}=\nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u),\;\;\;x\in \Omega,\;t>0,\\v_{t}=\triangle v-\beta v+\alpha u^{k_{1}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\0=\triangle w-\delta w+\gamma u^{k_{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\;\;w(x,0)=w_{0}(x), \;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega.\end{array}\right.$$
The system here is under a homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} >0, p\geq 2$. In addition, the function $f$ is smooth and satisfies that $f(s)\leq\kappa-\mu s^{l}$ for all $s\geq0$, with $\kappa\in \mathbb{R}, \mu>0, l>1$. It is shown that (i) if $l>\max\{ 2k_{1}, \frac{2k_{1}n}{2+n}+\frac{1}{p-1}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}-1, \frac{2k_{1}n}{2+n}+\frac{2-p}{p-1}\}$ with $l>2$, then system possesses a global bounded weak solution.

In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the P ötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than $\theta_{4}(t_1,T)$. The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.

In this paper, we are concerned with a three-dimensional non-isothermal model for the compressible nematic liquid crystal flows in a periodic domain. Under some smallness and structural assumptions imposed on the time-periodic force, we establish the existence of the time-periodic solutions to the system by using a regularized approximation scheme and the topological degree theory. We also prove a uniqueness result via energy estimates.

In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

This paper is concerned with the minimizers of $L^2$-subcritical constraint variational problems with spatially decaying nonlinearities in a bounded domain $\Omega$ of $\mathbb{R}^{N}$ ($N\ge 1$). We prove that the problem admits minimizers for any $M>0$. Moreover, the limiting behavior of minimizers as $M\to\infty$ is also analyzed rigorously.

In this paper, we are concerned with solutions to the fractional Schrödinger-Poisson system
$\begin{cases}(-\Delta)^su-\phi|u|^{2_s^*-3}u=\lambda u+\mu|u|^{q-2}u+|u|^{2_s^*-2}u,&x\in\mathbb R^3,\\(-\Delta)^s\phi=|u|^{2_s^*-1},&x\in\mathbb R^3,\end{cases} $
with prescribed mass $\int_{\mathbb{R}^3}|u|^2\mathrm{d}x = a^2,$ where $a>0$ is a prescribed number, $\mu>0$ is a paremeter, $s \in (0,1), 2<q<2^*_s$, and $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical Sobolev exponent. In the $L^2$-subcritical case, we show the existence of multiple normalized solutions by using the genus theory and the truncation technique; in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the doubly critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve upon some existing studies on the fractional Schrödinger-Poisson system with a nonlocal critical term.

This paper deals with the radial symmetry of positive solutions to the nonlocal problem
$$(-\Delta)^s_\gamma u =b(x) f(u)\quad {\rm in}\ \, B_1\setminus\{0\},\qquad u=h\quad {\rm in}\ \, \mathbb{R}^N\setminus B_1, $$
where $b\!:B_1\to\mathbb{R}$ is locally Hölder continuous, radially symmetric and decreasing in the $|x|$ direction, $f\!: \mathbb{R}\to\mathbb{R}$ is a Lipschitz function, $h\!:B_1\to\mathbb{R}$ is radially symmetric, decreasing with respect to $|x|$ in $\mathbb{R}^N\setminus B_1$, $B_1$ is the unit ball centered at the origin, and $(-\Delta)^s_\gamma $ is the weighted fractional Laplacian with $s\in(0,1), \gamma\in[0,2s)$ defined by
$$(-\Delta)_{\gamma}^s u(x) =c_{N,s}\lim_{\delta \rightarrow 0^{+}}\int_{\mathbb{R}^N\backslash{B_{\delta}(x)}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}|y|^{\gamma}{\rm d}y.$$
We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space
$$(-\Delta)^s_\gamma u(x)=b(x) f(u)\quad {\rm in}\ \, \mathbb{R}^N\setminus\{0\} ,$$
under suitable additional assumptions on $b$ and $f$. Our symmetry results are derived by the method of moving planes, where the main difficulty comes from the weighted fractional Laplacian. Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators
$$(-\Delta)^s u +\frac{\mu}{|x|^{2s}}u=b(x) f(u)\quad {\rm in}\ \, B_1\setminus\{0\},\qquad u=h\quad {\rm in}\ \, \mathbb{R}^N\setminus B_1,$$
under suitable additional assumptions on $b$, $f$ and $h$.

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations is studied by energy methods. Tilted perturbation refers to the fact that perturbations form an angle $\theta\in(0,\frac{\pi}{2})$ with the direction of the basic flows. By defining an energy functional, it is proven that plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbation when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free. In the case of stress-free boundaries, by taking advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals, it can be even shown that plane parallel shear flows are unconditionally nonlinearly exponentially stable for all Reynolds numbers, where the tilted perturbation can be either spanwise or streamwise.

In this paper, we consider the fully parabolic chemotaxis system with the general logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}u_t= \Delta(\gamma(v) u )+\lambda u-\mu u^{\kappa},~~~ &x \in \Omega, ~t>0,\\ v_t= \Delta v+wz, &x \in \Omega, ~t>0,\\w_t= -wz, &x \in \Omega, ~t>0,\\z_t= \Delta z - z+ u, &x\in \Omega, ~t>0,\\\end{array}\right.\end{eqnarray*}$
where $\Omega\subset \mathbb{R}^n (n\geq 1)$ is a smooth and bounded domain, $\lambda\geq 0, \mu\geq 0, \kappa>1$, and the motility function satisfies that $\gamma(v)\in C^3([0, \infty))$, $\gamma(v)>0$, $\gamma{'}(v)\leq0$ for all $v\geq 0$. Considering the Neumann boundary condition, we obtain the global boundedness of solutions if one of the following conditions holds: (i) $ \lambda=\mu=0, 1\leq n\leq 3; $(ii) $ \lambda> 0, \mu>0, ~\text{combined with}~ \kappa>1, 1\leq n\leq 3 ~~\text{or}~~\kappa>\frac{n+2}{4}, n>3. $ Moreover,we prove that the solution $(u, v, w, z)$ exponentially converges to the constant steady state $\left(\left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}, \frac{\int_{\Omega}v_0{\rm d}x+\int_{\Omega}w_0{\rm d}x}{|\Omega|}, 0, \left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}\right)$.

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular,are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal $L^2$ convergence rate of the $k$ ($\in [0, 3]$)-order spatial derivatives of the solution is $(1+t)^{-\frac{3+2k}{4}}$. Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.

In this paper, we mainly study the propagation properties of a nonlocal dispersal predator-prey system in a shifting environment. It is known that Choi et al. [J Differ Equ, 2021, 302: 807-853] studied the persistence or extinction of the prey and of the predator separately in various moving frames. In particular, they achieved a complete picture in the local diffusion case. However, the question of the persistence of the prey and of the predator in some intermediate moving frames in the nonlocal diffusion case was left open in Choi et al.'s paper. By using some a prior estimates, the Arzelà-Ascoli theorem and a diagonal extraction process, we can extend and improve the main results of Choi et al. to achieve a complete picture in the nonlocal diffusion case.

This paper deals with the problem of limit cycles for the whirling pendulum equation $\dot{x}=y,\ \dot{y}=\sin x(\cos x-r)$ under piecewise smooth perturbations of polynomials of $\cos x$, $\sin x$ and $y$ of degree $n$ with the switching line $x=0$. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations, which the generating functions of the associated first order Melnikov functions satisfy. Furthermore, the exact bound of a special case is given using the Chebyshev system. At the end, some numerical simulations are given to illustrate the existence of limit cycles.

Discrete feedback control was designed to stabilize an unstable hybrid neutral stochastic differential delay system (HNSDDS) under a highly nonlinear constraint %high nonlinearity, in the $H_{\infty}$ and exponential forms. Nevertheless, the existing work just adapted to autonomous cases, and the obtained results were mainly on exponential stabilization. In comparison with autonomous cases, non-autonomous systems are of great interest and represent an important challenge. Accordingly, discrete feedback control has here been adjusted with a time factor to stabilize an unstable non-autonomous HNSDDS, in which new Lyapunov-Krasovskii functionals and some novel technologies are adopted. It should be noted, in particular, that the stabilization can be achieved not only in the routine $H_{\infty}$ and exponential forms, but also the polynomial form and even a general form.

Drug resistance is one of the most intractable issues in targeted therapy for cancer diseases. It has also been demonstrated to be related to cancer heterogeneity, which promotes the emergence of treatment-refractory cancer cell populations. Focusing on how cancer cells develop resistance during the encounter with targeted drugs and the immune system, we propose a mathematical model for studying the dynamics of drug resistance in a conjoint heterogeneous tumor-immune setting. We analyze the local geometric properties of the equilibria of the model. Numerical simulations show that the selectively targeted removal of sensitive cancer cells may cause the initially heterogeneous population to become a more resistant population. Moreover, the decline of immune recruitment is a stronger determinant of cancer escape from immune surveillance or targeted therapy than the decay in immune predation strength. Sensitivity analysis of model parameters provides insight into the roles of the immune system combined with targeted therapy in determining treatment outcomes.