Consider a parabolic problem PDE, on a domain, say $(0,T)\times (x_1,x_2)$
Let's say we have
$Lu(x,t) = f(x,t;p)$
with boundary conditions $u(x_1,t)=g_1(t;p)$, $u(x_2,t)=g_2(t;p)$ and $f(x,0)=h(x;p)$
where $f,g_1,g_2,h$ are all $\alpha$-Holder continuous in $(x,t)$.
Hence solution exists for each value of alpha.
My question is, is it known if the solution $u(x,t;p)$ is continuously dependent on $p$ if $f,g_1,g_2,h$ are all continuous?