American Institute of Mathematical Sciences

$dF_n$/$dt=\varphi(F_n)(F_{n-1}-F_n)\quad\qquad\qquad (\star)$

where, for every $t, \{F_n(t), n=0,1,2,\ldots\}$ is a probability distribution function, and $\varphi$ is a positive function on $[0, 1]$. The equation $(\star)$ arose as a description of industrial economic development taking into accout processes of creation and propagation of new technologies. The paper contains a survey of the earlier received results including a multi-dimensional generalization and an application to the economic growth theory.
If $\varphi$ is decreasing then solutions of Cauchy problem for $(\star)$ approach to a family of wave-trains. We show that diffusion-wise asymptotic behavior takes place if $\varphi$ is increasing. For the nonmonotonic case a general hypothesis about asymtotic behavior is formulated and an analogue of a Weinberger's (1990) theorem is proved. It is argued that the equation can be considereded as an analogue of Burgers equation.

$z_t=-\alpha z\times (z\times z_{x x})+ z\times z_{x x}+z\times f(z), \qquad (\alpha \geq 0).$

The existence of unique smooth solutions is proved by using the technique of spatial difference and a priori estimates of higher-order derivatives in Sobolev spaces.

$ \int_M G(x, y)|\tau(h(y))|dV_y $ is bounded on each compact subset. $\qquad$ (1)

Here $\tau(h(x))$ is the tension field of the initial data $h(x)$.
Condition (1) is somewhat sharp as is shown by examples in the paper.

$ \partial_t u = (1 + i\nu)\Delta u + Ru- (1 + i\mu) |u|^2 u; \quad 0\le t < \infty, x\in\Omega $,

is investigated in a bounded domain ­$\Omega\subset \mathbb R^n$ with suffciently smooth boundary. Standard boundary conditions are considered: Dirichlet, Neumann or periodic. Existence and uniqueness of global smooth solutions is established for all real parameter values $\mu$ and $\nu$ if $n\le 2$, and for certain parameter values $\mu$ and $\nu$ if $n\ge 3$. Furthermore, dynamical properties of the CGL equation, such as existence of determining nodes, are shown. The proof of existence of smooth solutions hinges on the following inequality using the $L^2(\Omega)$-duality,

$|\mathfrak Im$ $<\Delta u ,\ |u|^{p-2}u>\le (|p-2|)/(2\sqrt{p-1})\mathfrak Re$ $< -\Delta u ,\ |u|^{p-2}u >.$

$u'(t) = Au(t) + \int_{-r}^0 k(s)A_1 u(s) ds + f(t),\quad t\ge 0;\quad u(t) = z(t), \quad t\in [-r,0]$

(where $A : D(A)\subset X \to X$ is a closed operator and $A_1 : D(A)\to X$ is continuous) is proved and applied to get a classical solution of the wave equation with memory effects

$ w_{t t} (t,x) = w_{x x}(t, x) + \int_{-r}^0 k(s) w_{x x} (t + s, x)ds + f(t, x), \quad t\ge 0,\quad x\in [0,l]$

To include also the Dirichlet boundary conditions and to get $C^2$-solutions, $D(A)$ is not supposed to be dense hence A is only a Hille-Yosida operator. The methods used are based on a reduction of the inhomogeneous equation to a homogeneous system of the first order and then on an immersion of $X$ in its extrapolation space, where the regularity and perturbation results of the classical semigroup theory can be applied.

$u_t - \Delta u = |u|^{p-1} u, \quad x\in\Omega,\quad t\in [0,T]$,

$u(t,x)=0, x\in\partial\Omega, \quad t\in [0,T] $,

$u(0,x) =u_0(x),\quad x\in\Omega $,

around a blow up point other than its centre of symmetry. We assume that $\Omega$ is a ball in $\mathbb R^N$ or $\Omega =\mathbb R^N$, and $p>1$. We show that $u$ behave as of a one-dimensional problem was concerned, that is, the possible asymptotic behaviors and final time profiles around an unfocused blow up point are the ones corresponding to the case of dimesion $N=1$.