A stability property of the stochastic heat equation (Q1910898)

The authors consider a sequence of one-dimensional nonlinear stochastic heat equations \[ {\partial u_n \over \partial t} (t,x) = {\partial^2 u_n \over \partial x^2} + f_n \bigl( u_n (t,x) \bigr) W_{t,x}, \quad u_n (0,x) = \varphi (x), \quad 0 < x < 1, \;t \geq 0, \tag{1} \] with homogeneous Neumann boundary conditions \(\partial u_n (t,x)/ \partial x = 0\), \(x = 0,1\), where \(W_{t,x}\) denotes a space-time white noise on a probability space \(\Omega\), \(E |\varphi (x) |^2\) is supposed to be bounded and the functions \(f_n : \mathbb{R} \to \mathbb{R}\) satisfy the uniform Lipschitz condition. Assuming that \(f_n \to f_0\) uniformly they prove that the mild solutions \(u_n\) of (1) converge to \(u_0\) in the norm of the space \(L^p (\Omega; {\mathcal C} ([0,t] \times [0,1]))\) for any \(p > 1\) and \(T \geq 0\).