In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the Navier-Stokes system with no external force in $\mathbb R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, for initial values $u_0$ small in a certain sense, we show that if the sequence of dilations $\lambda _n u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $\sqrt{t} u(t,\cdot\sqrt t)$ converges uniformly on $\mathbb R^N $ to ${{\boldsymbol{\mathcal S}}}(1)z$ along the subsequence $t_n=\lambda _n^2$, where ${{\boldsymbol{\mathcal S}}}(t)$ is the Navier-Stokes flow. If $N=2$ or $3$, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of an infinite-dimensional space. The resulting ``universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. Moreover, for a fixed $t_0>0$, ${{\boldsymbol{\mathcal S}}}(t_0)$ followed by an appropriate dilation generates a chaotic discrete dynamical system.

In this paper, we study a nonnegative blow-up solution of the Dirichlet problem for a quasilinear parabolic equation $(u^{\alpha})_t=\Delta u +f(u) + g(u(x_0(t),t))$ in $B(R)$, where $B(R)=\{ x \in \mathbf{R^N}\,;\, |x| < R\}$, $0 < \alpha \le 1$, $x_0(t)\in C^{\infty}([0,\infty) ;B(R))$ satisfies $x_0(t)\not = 0$, and $f(\xi)$ and $g(\xi)$ satisfy some blow-up conditions. We consider radial blow-up solutions $u(r,t)$ ($r=|x|$) which are non-increasing in $r$, and give the classification between total blow-up and single point blow-up according to the growth orders of $f$ and $g$. Especially in the case $\alpha =1$ we completely classify blow-up phenomena except for $f\sim g$ as follows. (I) When $g=o(f)$, any blow-up solution blows up only at the origin (single point blow-up); (II) When $f=o(g)$, (i) a single point blow-up solution exists (ii) there exists an initial data such that the solution blows up in the whole domain $B(R)$ (total blow-up) (iii) there are no other blow-up phenomena except total blow-up and single point blow-up.

In this paper we obtain for a generalized Tricomi operator fundamental solutions entirely supported in the hyperbolic region. Our method is based upon the notion of hypergeometric distributions introduced by S. Delache and J. Leray in [5].

We establish the existence of multiple solutions to the Dirichlet problem for the equation \[ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u \] on a bounded domain $\Omega$ of $\mathbb{R}^{N},$ $N\geq4.$ We show that, if $\lambda>0$ is not a Dirichlet eigenvalue of $-\Delta$ on $\Omega,$ this problem has at least $\frac{N+1}{2}$ pairs of nontrivial solutions. If $\lambda$ is an eigenvalue of multiplicity $m$ then it has at least $\frac{N+1-m}{2}$ pairs of nontrivial solutions.