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The inflow problem of full compressible NavierStokes equations is considered on the half-line (0,+\infty). First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state (0,+\infty) belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.

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.

No abstract uploaded!

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 $\ve\to0^+$.