American Institute of Mathematical Sciences

We consider the Brezis-Nirenberg problem

\begin{document}$ \begin{cases} \Delta_{{\mathbb S}^n}U +\lambda U + U^p = 0, \, U>0 & \text{in $\Omega_{\theta_1, \theta_2}$, }\\ U = 0&\text{on $\partial \Omega_{\theta_1, \theta_2}$, } \end{cases} $\end{document}

where \begin{document}$ \Omega_{\theta_1, \theta_2} $\end{document} is the set of the points whose great circle distance from \begin{document}$ (0, \ldots, 0, 1) $\end{document} is greater than \begin{document}$ \theta_2 $\end{document} and less than \begin{document}$ \theta_1 $\end{document}. If the annular domain is sufficiently thin, we show that the problem has a unique positive solution whose value depends only on the great circle distance from \begin{document}$ (0, \ldots, 0, 1) $\end{document} and there exists a nonradial bifurcation arising from the solution.

In this paper, we study some elliptic equation defined in \begin{document}$ \mathbb{R}^2 $\end{document} and involving a nonlinearity with new exponential growth condition including the doubly exponential growth at infinity. For that aim, we start by extending some new Trudinger-Moser type inequalities defined on the unit ball of different classes of weighted Sobolev spaces established by B. Ruf and M. Calanchi to the whole space \begin{document}$ \mathbb{R}^2. $\end{document}

We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent \begin{document}$ \alpha $\end{document}. For sign-changing solutions, the case \begin{document}$ \alpha = 0 $\end{document} -Lane-Emden equation- is included. The obtained solutions form global continua which branch off from the curve of radial solutions \begin{document}$ p\mapsto u_p $\end{document}, and the number of branching points increases with both the number of nodal zones and the exponent \begin{document}$ \alpha $\end{document}. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between different branches, thus allowing to separate them in some cases.

In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution \begin{document}$ u $\end{document} of the multi-dimensional semilinear Tricomi equation

\begin{document}$ \begin{equation*} \partial_t^2 u-t\, \Delta u = |u|^p, \quad \big(u(0, \cdot), \partial_t u(0, \cdot)\big) = (u_0, u_1), \end{equation*} $\end{document}

where \begin{document}$ t>0 $\end{document}, \begin{document}$ x\in \mathbb R^n $\end{document}, \begin{document}$n\geq2$\end{document}, \begin{document}$ p>1 $\end{document}, and \begin{document}$ u_i\in C_0^{\infty}( \mathbb R^n) $\end{document} (\begin{document}$ i = 0, 1 $\end{document}). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of \begin{document}$\|u(t, \cdot)\|_{L^\infty_x(\mathbb R)}$\end{document} is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of \begin{document}$u$\end{document} in \begin{document}$t$\end{document}, then the crucial weighted Strichartz estimates are established and the global existence of solution \begin{document}$ u $\end{document} is proved when \begin{document}$ p>5 $\end{document}.

This paper investigates prey-taxis system with rotational flux terms

\begin{document}$ \begin{equation*} \begin{cases} u_t = \Delta u-\nabla\cdot(uS(x,u,v)\nabla v)+uv-\rho u,& x\in\Omega,\ t>0,\\ v_t = \Delta v-\xi uv+\mu v(1-\alpha v),&x\in\Omega,\ t>0, \end{cases} \end{equation*} $\end{document}

under no-flux boundary conditions in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n\; (n\geq1) $\end{document} with smooth boundary. Here the matrix-valued function \begin{document}$ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{n\times n}) $\end{document} fulfills \begin{document}$ |S(x,u,v)|\leq\frac{S_0(v)}{(1+u)^\theta}(\theta\geq0) $\end{document} for all \begin{document}$ (x,u,v)\in\bar{\Omega}\times[0,\infty)^2 $\end{document} with some nondecreasing function \begin{document}$ S_0 $\end{document}. It is proved that for nonnegative initial data \begin{document}$ u_0\in C^0(\overline{\Omega}) $\end{document} and \begin{document}$ v_0\in W^{1,q}(\Omega) $\end{document} with some \begin{document}$ q>\max\{n,2\} $\end{document}, if one of the following assumptions holds: (i) \begin{document}$ n = 1 $\end{document}, (ii) \begin{document}$ n\geq2, \theta = 0 $\end{document} and \begin{document}$ S_0(m)m<\frac{2}{\sqrt{3n(11n+2)}} $\end{document}, (iii) \begin{document}$ \theta>0 $\end{document}, then the model possesses a global classical solution that is uniformly bounded. Where \begin{document}$ m: = \max\{\|v_0\|_{L^\infty(\Omega)}, \frac{1}{\alpha}\} $\end{document}.

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of \begin{document}$ \pi/2 $\end{document}.

We discuss the stability problem for binary mixtures systems coupled with heat equations. The present manuscript covers the non-classical thermoelastic theories of Coleman-Gurtin and Gurtin-Pipkin - both theories overcome the property of infinite propagation speed (Fourier's law property). We first state the well-posedness and our main result is related to long-time behavior. More precisely, we show, under suitable hypotheses on the physical parameters, that the corresponding solution is stabilized to zero with exponential or rational rates.

We establish the global existence of subsonic solutions to a two dimensional Riemann problem governed by a self-similar pressure gradient system for shock diffraction by a convex cornered wedge. Since the boundary of the subsonic region consists of a transonic shock and a part of a sonic circle, the governing equation becomes a free boundary problem for nonlinear degenerate elliptic equation of second order with a degenerate oblique derivative boundary condition. We also obtain the optimal \begin{document}$ C^{0,1} $\end{document}-regularity of the solutions across the degenerate sonic boundary.

In this paper, we use the Perron method to prove the existence and uniqueness of the exterior problem for a kind of parabolic Monge-Ampère equation \begin{document}$ -u_t+\log\det D^2u = f(x) $\end{document} with prescribed asymptotic behavior at infinity, where \begin{document}$ f $\end{document} is asymptotically close to a radial function at infinity. We generalize the results of both the elliptic exterior problems and the parabolic interior problems for the Monge-Ampère equations.

We perform an weighted Sobolev space approach to prove a Trudinger-Moser type inequality in the upper half-space. As applications, we derive some existence and multiplicity results for the problem

\begin{document}$ \begin{cases} -\Delta u+h(x)|u|^{q-2}u = a(x) f(u), &\mbox{in } \mathbb{R}^2_+,\\ \dfrac{\partial u}{\partial \nu}+u = 0, &\mbox{on } \partial\mathbb{R}^2_+, \end{cases} $\end{document}

under some technical condition on \begin{document}$ a $\end{document}, \begin{document}$ b $\end{document} and the the exponential nonlinearity \begin{document}$ f $\end{document}. The ideas can also be used to deal with Neumann boundary conditions.

In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.

This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding Du's et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} when the initial density and the initial magnetic field are decay not too show at infinity, and Ji's et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.

In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain \begin{document}$ \Omega $\end{document} of \begin{document}$ \mathbb{R}^N $\end{document}. Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.

We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.