In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.

Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2\Delta u_1+\lambda_1u_1=\mu_1u_1^3+\alpha_1u_1^{p-1}+\beta u_2^2u_1\quad&\text{in }\Omega,\\ &-\ve^2\Delta u_2+\lambda_2u_2=\mu_2u_2^3+\alpha_2u_2^{p-1}+\beta u_1^2u_2\quad&\text{in }\Omega,\\ &u_1,u_2>0\quad\text{in }\Omega,\quad u_1=u_2=0\quad\text{on }\partial\Omega,\endaligned\right. \end{equation*} where $\Omega\subset\bbr^4$ is a bounded domain, $\lambda_i,\mu_i,\alpha_i>0$ $(i=1,2)$ and $\beta\not=0$ are constants, $\ve>0$ is a small parameter and $2<p<2^*=4$. By using variational methods, we study the existence of ground state solutions to this system for sufficiently small $\ve>0$. The concentration behaviors of least-energy solutions as $\ve\to0^+$ are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as $\ve\to0^+$.

In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier-Stokes equations is proved with the initial date being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many literatures concerning the well-posedness of compressible Navier-Stokes equations in the presence of vacuum.

where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,

During the past year a group of 12 graduate students, a visiting researcher and a postdoctoral fellow, supported in part by this contract, have studied various nonlinear partial differential equations. Seminars were held twice weekly on the topic of soliton theory. The graduate students involved have been studying the problem of singularity that arises in nonlinear equations.