Let $\lambda \colon [0, T] \times (0, \infty) \to [0, \infty)$ as well as some reasonable $f \colon (0, \infty) \to [0, \infty)$ and consider the following Cauchy boundary value problem: $$\partial_t u(t, x) = \frac{1}{2}\lambda^2(t, x) x^2 \partial_x^2 u(t,x),\quad u(0, \cdot) = f.$$
I am interested in identifying the boundary value problem as a continuous operator between normed spaces. Are there normed spaces $X$ and $Y$ such that it is safe to take $\lambda$ from $X$ and find the corresponding solution $u_\lambda$ to be existing in $Y$, and that all in a way that makes $$X \to Y, \quad \lambda \mapsto u_\lambda$$ continuous?