We study the Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation, several profiles of blowup solution have been noticed. Here, we study blowup in infinite time in accordance with the residual vanishing and Hamiltonian control on collapse dynamics.

In this paper, we study the hydrodynamic limit of a binary mixture of rigid spheres. When Knudsen numbers of two different species are equal and go to zero, we show formally that the hydrodynamic variables satisfy the compressible Euler and Navier-Stokes equations. Like single species gas, we develop Enskog-Chapman theory up to the second order. It turns out that the macro velocities corresponding to the different spheres are equal and the ratio of the temperatures is the mass ratio of the spheres. However, the macro mass densities satisfy independent conservation laws in the case of the compressible flows. Explicit formulas of viscosity, heat conductivity and heat diffusion coefficient are established in term of particle parameters.

Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations (diffusion-ODE models), which differ from the usual reaction-diffusion systems. One of the mechanisms of pattern formation in such systems is based on the existence of multiple steady states and hysteresis in the ODE subsystem. Diffusion tries to average different states and is the cause of spatio-temporal patterns. In this paper, we provide a systematic description of stationary solutions of such systems having the form of transition or boundary layers. The solutions are discontinuous in the case of non-diffusing variables whose quasi-stationary dynamics exhibit hysteresis. The considered model is motivated by biological applications and elucidates a possible mechanism of formation of patterns with sharp transitions.

The ability of eukaryotic cells to actively move along different substrates plays a vital role in many biological processes. A key player in these processes is the cytoskeleton. In [7], we introduced a minimal hyperbolic-parabolic model for the reorganization of the actin cytoskeleton of a generic cell resting on a flat substrate and turning into a polarized state upon some external cue. In this paper, we derive moving boundary conditions for that model and by this allow for the description of actual motion. For the free boundary problem we prove short time well-posedness for a wide class of initial conditions and analyze the emergence of Dirac measures in the densities of actin filament tips. These have a biophysical interpretation as sharp polymerization fronts which are experimentally observed in [19], for example. Further, simulations will illustrate the motion of an initially symmetric resting cell and the emergence of sharp actin fronts from initially smooth distributions.

In this paper, we study the problem \begin{equation} \left\{\begin{array}{ll} -\Delta u=|x|^\alpha u^{p_\alpha}+{\epsilon}|x|^{{\beta}}u & \hbox{ in }\Omega\\ u>0 & \hbox{ in }\Omega\\ u=0 & \hbox{ on }{\partial}\Omega, \end{array}\right. \end{equation} where $p_\alpha=\frac{N+2+2\alpha}{N-2}$, $\Omega$ is a smooth bounded domain of ${\mathbb R}^N$ with $0\in\Omega$ and $N\ge4$. We show that, for $\alpha\ge0$ and $0\le{\beta}\le N-4$, there exists one solution concentrating at $x=0$ as ${\epsilon}\rightarrow0$. Moreover, we prove that, if $\Omega$ is a ball, there exist no radial solution if $\alpha={\beta}>N-4$.

We establish the time local well-posedness of a multi-component parabolic-elliptic drift-diffusion model in two space dimensions in a scaling critical weighted space. This is a generalization of former result Kurokiba-Ogawa [20] and corresponds to the critical case for multiple system treated in Kurokiba [18]. In order to show the critical well-posedness, we introduce a refined version of Brezis-Gallouët inequality in term of the Besov space and apply it to the some weighted estimate in two dimensions.

We consider the behavior of solutions to the Cauchy problem for a semilinear heat equation with supercritical nonlinearity. We study the convergence of solutions to steady states in a weighted norm, and show the global attractivity property of steady states. We also give its convergence rate for a class of initial data. Proofs are given by a comparison method based on matched asymptotic expansion.

We consider the biharmonic Liouville-Gel'fand problem under the Navier boundary condition in four space dimension. Under the nondegeneracy assumption of blow up points of multiple blowing-up solutions, we prove several estimates for the linearized equations and obtain some convergence result. The result can be seen as a weaker version of the asymptotic nondegeneracy of multi-bubble solutions, which was recently established by Grossi-Ohtsuka-Suzuki in two-dimensional Laplacian case.

We study the critical problem \begin{equation} \left\{ \begin{array}{ll} \Delta ^{2}u=u^{\frac{N+4}{N-4} } & \mbox{ in }\Omega\setminus \overline{B(\xi_0,\varepsilon) }, \\ u>0 & \mbox{ in }\Omega\setminus \overline{B(\xi_0,\varepsilon) }, \\ u=\Delta u=0 & \mbox{ on }\partial (\Omega \setminus \overline{B(\xi_0,\varepsilon) }), \end{array} \right. \tag{P$_\varepsilon$} \end{equation} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N\ge5$, $\xi_0\in\Omega$ and $B(\xi_0,\varepsilon)$ is the ball centered at $\xi_0$ with radius $\varepsilon>0$ small enough. We construct solutions of (P$_\varepsilon$) blowing-up at the center of the hole as the size of the hole goes to zero.