We consider the degenerate Keller-Segel system (KS) of Nagai type below. We prove that when $m > 2-\frac{2}{N}$, the problem (KS) is solvable globally in time without any restriction on the size of the initial data and that when $1 < m \le 2-\frac{2}{N}$, the problem (KS) evolves in a finite time blow-up for some large initial data. Hence, we completely classify the existence and non-existence of the time global solution by means of the exponent $m=2-\frac{2}{N}$, which generalizes the Fujita exponent for (KS).

We consider the scattering problem for two types of nonlinear Klein-Gordon equations. One is the equation of the Hartree type, and the other one is the equation with power nonlinearity. We show that the scattering operator for the equation of the Hartree type converges to that for the one with power nonlinearity in some sense. Our proof is based on some inequalities in the Lorentz space, and a strong limit of Riesz potentials.

We study the initial-boundary-value problem for the fractional heat equation on a segment $ ( 0,a ) $ \begin{equation} \left \{ \begin{array}{c} u_{t}+\lambda \vert u \vert ^{\rho }\text{ }u+C_{\alpha }\partial _{x}^{\alpha }u=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right . \tag*{(0.1)} \end{equation} where $\lambda \in R$ , $\rho \geq 0$ , $\alpha \in ( 1,\frac{3}{2} ] $ $,$ the constant $C_{\alpha }$ is chosen by a dissipative condition, such that $\ \text{Re}C_{\alpha }p^{ [ \alpha ] +1- \{ \alpha \} }>0$ for $\text{Re}p=0$ and \begin{equation*} \partial _{x}^{\alpha }u=\int_{0}^{x}\frac{\partial _{s}^{ [ \alpha ] +1}u(s,t)}{ ( x-s ) ^{ \{ \alpha \} }}ds. \end{equation*} Here $ [ \alpha ] $ and $ \{ \alpha \} $ are integer and fractional parts of $\alpha .$ The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem (0.1) and to find the main term of the asymptotic representation of solutions.

In this paper we study the initial-boundary-value problem for the magnetohydrodynamic system in three-dimensional exterior domains. We show an existence theorem of global in time strong solutions for small $L^3$-initial data and we also show its asymptotic behavior when time goes to infinity.

We study some aspects of the local behaviour of infinity-harmonic functions. We study one-sided radial derivatives at points of extrema on the boundary of concentric balls and relate it to some issues in differentiability. In particular we show differentiability of these functions at such points of extrema. We use convexity and the Harnack inequality to obtain our results.