Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation: \begin{equation}\label{e1} \begin{cases} \dot y(x,t)= y_{xx}(x,t)+ u(x,t)y(x,t) \quad \text{in} \ \Omega \times (0,T), \\ y=0 \ \text{on} \ \Gamma \times (0,T), \\ y(0)=y_0 \ \text{in} \ \Omega \end{cases} \end{equation} where $u\in L^2(\Omega \times (0,T))$ is a given function. The objective is to prove that the map $u \mapsto y$ is weak-norm continuous.
We consider a sequence of functions $u_n$ that weakly converges to $u^*$, and we let $y_n$ and $y^*$ be the solutions of the equation above corresponding to $u_n$ and $u^*$, respectively.
My question is: How to demonstrate that $y^n$ converges to $y^*$ in the $L^2$ norm?