Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds (Q6606149)

Nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain, driven by a Gaussian noise that is white in time and colored in space, are considered by the authors, with either Dirichlet or Neumann boundary conditions. The existence, uniqueness, and moment bounds of the random field solution are established under measure-valued initial data \(\nu\). The two-point correlation function of the solution is also studied, with explicit upper and lower bounds obtained. For \(C^{1,\alpha}\)-domains with Dirichlet boundary conditions, the requirement for the initial data \(\nu\) to be a finite measure is relaxed, and the moment bounds are improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to \(|\nu|\). As an application, the authors demonstrate that the solution becomes fully intermittent for sufficiently high levels \(\lambda\) of noise under the Dirichlet condition, and for all \(\lambda > 0\) under the Neumann condition.