American Institute of Mathematical Sciences

In this paper, we consider a two-group SIR epidemic model with bilinear incidence in a patchy environment. It is assumed that the infectious disease has a fixed latent period and spreads between two groups. Firstly, when the basic reproduction number \begin{document}$ \mathcal{R}_{0}>1 $\end{document} and speed \begin{document}$ c>c^{\ast} $\end{document}, we prove that the system admits a nontrivial traveling wave solution, where \begin{document}$ c^{\ast} $\end{document} is the minimal wave speed. Next, when \begin{document}$ \mathcal{R}_{0}\leq1 $\end{document} and \begin{document}$ c>0 $\end{document}, or \begin{document}$ \mathcal{R}_{0}>1 $\end{document} and \begin{document}$ c\in(0,c^{*}) $\end{document}, we also show that there is no positive traveling wave solution, where \begin{document}$ k = 1,2 $\end{document}. Finally, we discuss and simulate the dependence of the minimum speed \begin{document}$ c^{\ast} $\end{document} on the parameters.

We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, \begin{document}$ M(6)\ge 48. $\end{document}

In this paper, we consider the following Schrödinger equation with singular potential:

\begin{document}$ \begin{equation*} \begin{aligned} -\Delta u + V(|x|)u = f(u),\ \, x\in \mathbb{R}^{N}, \end{aligned} \end{equation*} $\end{document}

where \begin{document}$ N\geqslant 3 $\end{document}, \begin{document}$ V $\end{document} is a singular potential with parameter \begin{document}$ \alpha\in(0,2)\cup(2,\infty) $\end{document}, the nonlinearity \begin{document}$ f $\end{document} involving critical exponent. First, by using the refined Sobolev inequality, we establish a Lions-type theorem. Second, applying Lions-type theorem and variational methods, we show the existence of ground state solution for above equation. Our result partially extends the results in Badiale-Rolando [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006)], and Su-Wang-Willem [Commun. Contemp. Math. 9 (2007)].

Our purpose in this paper is to classify the non-topological solutions of equations

\begin{document}$ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E) $\end{document}

where \begin{document}$ \{\delta_{p_i}\}_{i = 1}^k $\end{document} (resp. \begin{document}$ \{\delta_{q_j}\}_{j = 1}^l $\end{document}) are Dirac masses concentrated at the points \begin{document}$ \{p_i\}_{i = 1}^k $\end{document}, (resp. \begin{document}$ \{q_j\}_{j = 1}^l $\end{document}), \begin{document}$ n_i $\end{document} and \begin{document}$ m_j $\end{document} are positive integers. Denote \begin{document}$ N = \sum^k_{i = 1}n_i $\end{document} and \begin{document}$ M = \sum^l_{j = 1}m_j $\end{document} satisfying that \begin{document}$ N-M>1 $\end{document}.

Problem \begin{document}$ (E) $\end{document} arises from gauged sigma models and we first construct an extremal non-topological solution \begin{document}$ u $\end{document} of \begin{document}$ (E) $\end{document} with asymptotic behavior

\begin{document}$ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $\end{document}

and with total magnetic flux \begin{document}$ 4\pi (N-M-1) $\end{document}. And then we do the classification for non-topological solutions of \begin{document}$ (E) $\end{document} with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.

A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter \begin{document}$ \lambda $\end{document} which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for \begin{document}$ \lambda $\end{document} sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.

We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth.

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equation*} $\end{document}

where \begin{document}$ \lambda > 0, N \geq 3, \alpha \in (0, N) $\end{document}. The potential \begin{document}$ V $\end{document} is a continuous function and \begin{document}$ I_\alpha $\end{document} denotes the standard Riesz potential. Assume also that \begin{document}$ 1 < q < 2 $\end{document}, \begin{document}$ 2_\alpha < p < 2^*_\alpha $\end{document} where \begin{document}$ 2_\alpha = (N+\alpha)/N $\end{document}, \begin{document}$ 2_\alpha = (N+\alpha)/(N-2) $\end{document}. Our main contribution is to consider a specific condition on the parameter \begin{document}$ \lambda > 0 $\end{document} taking into account the nonlinear Rayleigh quotient. More precisely, there exists \begin{document}$ \lambda^* > 0 $\end{document} such that our main problem admits at least two positive solutions for each \begin{document}$ \lambda \in (0, \lambda^*] $\end{document}. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter \begin{document}$ \lambda^*> 0 $\end{document} is optimal in some sense which allow us to apply the Nehari method.

We develop highly anisotropic Carleson measure over multi-level ellipsoid covers \begin{document}$ \Theta $\end{document} of \begin{document}$ \mathbb{R}^n $\end{document} that are highly anisotropic in the sense that the ellipsoids can change rapidly from level to level and from point to point. Then we show that the Carleson measure \begin{document}$ \mu $\end{document} is sufficient for which the integral defines a bounded operator from \begin{document}$ H^p(\Theta) $\end{document} to \begin{document}$ L^p(\mathbb{R}^{n+1}, \, \mu),\ 0<p\leq 1 $\end{document}. Finally, we give several equivalent Carleson measures adapted to multi-level ellipsoid covers and obtain a specific Carleson measure induced by the highly anisotropic BMO functions.

We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.

We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.

In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number \begin{document}$ \mathscr{R}_{0} $\end{document} which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.

This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.

We study a kinetic-fluid model in a \begin{document}$ 3D $\end{document} bounded domain. More precisely, this model is a coupling of the Vlasov-Fokker-Planck equation with the local alignment force and the compressible Navier-Stokes equations with nonhomogeneous Dirichlet boundary condition. We prove the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient \begin{document}$ \gamma> 3/2 $\end{document}) and hence extend the existence result of Choi and Jung [Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, arXiv: 1912.13134v2], where the velocity of the fluid is supplemented with homogeneous Dirichlet boundary condition.

We consider a general inhomogeneous parabolic initial-boundary value problem for a \begin{document}$ 2b $\end{document}-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers \begin{document}$ s $\end{document} and \begin{document}$ s/(2b) $\end{document} and with a function \begin{document}$ \varphi:[1,\infty)\to(0,\infty) $\end{document} that varies slowly at infinity. The function parameter \begin{document}$ \varphi $\end{document} characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.

This note studies the asymptotics of radial global solutions to the non-linear fractional Schrödinger equation

\begin{document}$ i\dot u-(-\Delta)^s u+|u|^{p-2}(I_\alpha *|u|^p)u = 0. $\end{document}

Indeed, using a new method due to Dodson-Murphy [10], one proves that, in the inter-critical regime, under the ground state threshold, the radial global solutions scatter in the energy space.

We consider the conserved phase-field system

\begin{document}$\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$\end{document}

where \begin{document}$ \tau>0 $\end{document} is a relaxation time, \begin{document}$ \delta>0 $\end{document} is the viscosity parameter, \begin{document}$ \epsilon\in (0,1] $\end{document} is the heat capacity, \begin{document}$ \phi $\end{document} is the order parameter, \begin{document}$ u $\end{document} is the absolute temperature, the Laplace operator \begin{document}$ N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega) $\end{document} is subject to either Neumann boundary conditions (in which case \begin{document}$ \Omega\subset{\mathbb R}^d $\end{document} is a bounded domain with smooth boundary) or periodic boundary conditions (in which case \begin{document}$ \Omega = \Pi_{i = 1}^d(0,L_i), $\end{document} \begin{document}$ L_i>0 $\end{document}), \begin{document}$ d = 1,2 $\end{document} or 3, and \begin{document}$ G(\phi) = \int_0^\phi g(\sigma)d\sigma $\end{document} is a double-well potential. Let \begin{document}$ j = 1 $\end{document} when \begin{document}$ d = 1 $\end{document} and \begin{document}$ j = 2 $\end{document} when \begin{document}$ d = 2 $\end{document} or 3. We assume that \begin{document}$ g\in{\mathcal C}^{j+1}(\mathbb R) $\end{document} and satisfies the conditions \begin{document}$ g'(\phi)\geq -{\mathscr C}_1 $\end{document}, \begin{document}$ G(\phi)\ge -{\mathscr C}_2 $\end{document} and \begin{document}$ (\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi)) $\end{document} (\begin{document}$ {\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho) $\end{document}, \begin{document}$ l = 3,4 $\end{document}, whenever \begin{document}$ |m(\phi)|\le \varrho $\end{document}), where \begin{document}$ \varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0 $\end{document}, \begin{document}$ {\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6>0 $\end{document} and \begin{document}$ m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx $\end{document}. For instance, \begin{document}$ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $\end{document} \begin{document}$ p\in{\mathbb N}, $\end{document} \begin{document}$ p\ge 2, $\end{document} \begin{document}$ a_{2p-1}>0, $\end{document} satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space \begin{document}$ {\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2}) $\end{document} equipped with the norm \begin{document}$ \|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2} $\end{document}. Moreover, we demonstrate that the global attractor is upper semicontinuous at \begin{document}$ \epsilon = 0 $\end{document} in the metric induced by the norm \begin{document}$ \|.\|_{{\mathcal V}_{j+1}} $\end{document}. In addition, the exponential attractors are proven to be Hölder continuous at \begin{document}$ \epsilon = 0 $\end{document} in the metric induced by the norm \begin{document}$ \|.\|_{{\mathcal V}_{j}} $\end{document}. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition \begin{document}$ |g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right), $\end{document} \begin{document}$ {\mathscr C}_7>0 $\end{document}, \begin{document}$ p>0 $\end{document} is arbitrary when \begin{document}$ d = 1, 2 $\end{document} and \begin{document}$ p\in [0,3] $\end{document} when \begin{document}$ d = 3 $\end{document}, was required, preventing \begin{document}$ g $\end{document} to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.