A stochastic Stefan problem
The present article is concerned with a stochastic perturbation of the Stefan problem. More precisely, the authors deal with the stochastic partial differential equation \[ \begin{aligned} \frac{\partial u}{\partial t} (t,x) &= \frac{\partial^2 u}{\partial x^2}(t,x) + \alpha u(t,x) + u(t,x) d\zeta_t(x), \quad x > \beta(t),\\ \lim_{x \downarrow \beta(t)} \frac{\partial u}{\partial x}(t,x) &= - \rho \dot{\beta}(t),\\ u(0,x) &= u_0(x), \quad x \in \mathbb{R},\end{aligned} \] on \[ \{ (t,x) \in \mathbb{R}_+ \times \mathbb{R} \mid u(t,x) > 0 \} = \{ (t,x) \in \mathbb{R}_+ \times \mathbb{R} \mid x > \beta(t) \}. \] \(\beta\) determines the boundary, \(\zeta\) is a Gaussian field of the form \[ \zeta_t(x) = \int_{s=0}^t \int_{y \in \mathbb{R}} \eta(x-y) W(ds,dy) \] with a Brownian sheet \(W\), and solutions are understood in a weak sense. The authors prove the existence of weak solutions \(\{ u(t,\cdot) \mid 0 \leq t < \tau \}\) and show that the domain of existence is given by \[ \tau \leq \inf \bigg\{ t \geq 0 : \bigg| \frac{\partial u}{\partial x} (t-,\beta(t)) \bigg| = \infty \bigg\}. \] A theorem regarding uniqueness of solutions is provided as well.
stochastic partial differential equation
scientific article; zbMATH DE number 6124637