Santiago Juan Saglietti
Universita' di Buenos Aires
Titolo: Metastability for small random perturbations of a PDE with blow-up
Venerdi' 17 Aprile 2015 ORE 14:30
Dipartimento di Matematica e Fisica
Universita' degli Studi Roma Tre
AULA 311 (SEMINARI) Largo San L. Murialdo,1
Abstract
We consider the stochastic PDE $u_t = u_{xx} + u^p + varepsilon
dot{W}$ with homogenous Dirichlet boundary conditions, where $p > 1$,
$varepsilon > 0$ is a small fixed parameter and $dot{W}$ stands for
space-time white noise. It is well known that the associated
deterministic PDE (i.e. $varepsilon = 0$ in the equation above) admits
exactly one asymptotically stable equilibrium and a countable family
of unstable equilibria with increasing energy. Furthermore, for
certain initial conditions it can be shown that the solution of the
deterministic PDE explodes in finite time. We show that, for $p < 5$
and initial conditions in the domain of attraction of the
asymptotically stable equilibrium, the solution $X_varepsilon$ of the
SPDE satisfies in the limit as $varepsilon$ tends to zero the
classical description of metastability featured in [1]: the averages
of $X_varepsilon$ remain stable and close to the equilibrium up until
the explosion time which, when suitably rescaled, converges in
distribution to an exponential random variable. Furthermore, for
certain initial conditions in the domain of explosion (and any value
of $p > 1$) we show the continuity of the explosion time as
$varepsilon$ tends to zero.