We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.

A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to investigate a stochastic porous medium equation.

The well-known theory of the “rational canonical form of an operator” describes the invariant factors, or equivalently, elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space V over a given field F. A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The ...