We consider the degenerate Keller-Segel system (KS)$_m$ with $1 < m \le 2-\frac{2}{N}$ of Nagai type below. Our aim is to find the two types of conditions on the initial data which divide the situation of the solution $(u,v)$ into the global existence and the finite time blow-up; one is the assumption on the size of $R_m(u_0):=\displaystyle \frac{\int u_0 v_{0} \ dx}{\int u_0^{m} \ dx}$; the other one is the assumption on the size of $\int u_0^{\frac{N(2-m)}{2}} \ dx$. Moreover, we discuss the critical case of $m=2-\frac{2}{N} \ (N\ge3)$. We find the upper bound and the lower bound on the size of the $L^1(=L^{\frac{N(2-m)}{2}})$-norm of the initial data which assures the global existence and the finite time blow-up, respectively. Our results cover the well-known threshold number $\frac{8\pi}{\alpha \chi}$ in $\mathbb R^2$ for the semi-linear case, since both bounds correspond to $\frac{8\pi}{\alpha \chi}$ by substituting $m=1$ and $N=2$ formally. For our main results, we also give a proof of the mass conservation law in $\mathbb R^N$ to (KS)$_m$.

We study the positive solution set of the following quasilinear elliptic system: $$ \begin{cases} \Delta u+u(a-u-cv)=0 \ \ & \mbox{in} \ \ \Omega,\\ \Delta \Big [ \Big ( \mu+\dfrac{1}{1+\beta u} \Big ) v \Big ]+v(b+du-v)=0 \ \ & \mbox{in} \ \ \Omega,\\ u=v=0 \ \ & \mbox{on} \ \ \partial\Omega, \end{cases} $$ where $\Omega $ is a bounded domain in $\boldsymbol{R}^{N}$, $a, b, c, d$, and $\mu $ are positive constants, and $\beta $ is a nonnegative constant. This system is the stationary problem associated with a prey-predator model with the strongly coupled diffusion $\Delta (\frac{v}{1+\beta u})$, and $u$ (respectively $v$) denotes the population density of the prey (respectively the predator). In the previous paper by Kadota and Kuto \cite{KK}, we obtained the bifurcation branch of the positive solutions, which extends globally with respect to the bifurcation parameter $a$. In the present paper, we aim to derive the nonlinear effect of large $\beta $ on the positive solution continuum. We obtain two {\it shadow systems} in the limiting case as $\beta\to\infty $. From the analysis for the shadow systems, we prove that in the large $\beta $ case, the positive solutions satisfy $\| u\|_{\infty }=O(1/\beta)$ if $a$ is less than a threshold number, while the positive solutions can be approximated by a positive solution of the associated system without the strongly coupled diffusion if $a$ is large enough.

In this paper we consider the following elliptic equation: \begin{equation*} \begin{cases} \; -\Delta u + u = f(x,u) \qquad \text{in}\; \mathbf{R}^N, \\ \quad u \in H^1(\mathbf{R}^N). \end{cases} \end{equation*} Here $f(x,u)$ is invariant under a finite group action $G \subset O(N)$ which acts on $S^{N-1}$ effectively. When $N \geq 3$, we show the existence of a positive solution without global assumptions like: $ u \mapsto \frac{f(x,u)}{u} $ is increasing in $ u > 0$, $f(x,u) \geq f^{\infty}(u)$. We can deal with asymptotically linear equations as well as superlinear equations. Interaction estimates between solution at infinity play important roles in our argument.

We study a nonlinear hyperbolic system of balance laws that arises from an entropy-based moment closure of transfer radiative equations, and that involves a small parameter $\epsilon$. Under physical assumptions, global existence for Cauchy problems with smooth and small data is established through the energy method.

An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial-boundary-value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval.