Abstract:
Under which conditions on a holomorphic germ $u(x)$ at a point
$x_0\in\mathbb C$ can one find a holomorphic germ $u(x,t)$ at a point
$(x_0,t_0)\in\mathbb C^2$ satsifying an evolution equation $u_t=F[u]$ in
a neighbourhood of this point (where $F[u]$ is a prescribed polynomial
in $u$ and its partial derivatives with respect to $x$) and the initial
condition $u(x,t_0)=u_0(x)$? We shall present classical results (surprising
and not well-known) that completely answer this question in the case when
the differential polynomial $F[u]$ is linear, as well as some recent
advances in this problem.
|
