Let's say we have a parabolic PDE system:
$$ (PDE) \hspace{0.5cm} u_t+f(u)_x=\mu \cdot u_{xx}, $$ where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, \: n\geq 2$. And let's consider the different initial conditions:
$$(IC1) \hspace{0.5cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} $$
$$(IC2) \hspace{0.5cm} u(x,0)= \begin{cases} 0, x<-a \\[2ex] u_l, -a<x<0 \\[2ex] u_r, 0<x<a\\[2ex] 0, x>a \end{cases} $$
$$(IC3) \hspace{0.5cm} u(x,0)= \begin{cases} 0, x<-a-\sigma\\[2ex] smooth \: function_1, -a-\sigma<x<-a+\sigma \\[2ex] u_l, -a+\sigma<x<-\sigma \\[2ex] smooth \: function_2, -\sigma<x<\sigma \\[2ex] u_r, \sigma<x<a-\sigma \\[2ex] smooth \: function_3, a-\sigma<x<a+\sigma \\[2ex] 0, x>a+\sigma \end{cases} $$ where $u_l$ and $u_r$ are constants. Initial conditions $(IC1)$ are the usual Riemann conditions. Initial conditions $(IC2)$ are derived from $(IC1)$ by applying on it a cut off function that is equal to 1 on $[-a,a]$ and zero elsewhere. Conditions $(IC3)$ are derived from $(IC2)$ by using convolution with a moliffier $\varphi_{\sigma}$.
As far as I know solutions of the problems $(PDE)-(IC1)$, $(PDE)-(IC2)$, $(PDE)-(IC3)$ exist. And we see that when $\sigma \rightarrow 0$, $(IC3)$ converge to $(IC2)$, and when $a \rightarrow\infty$, $(IC2)$ converge to $(IC1)$.
I am interested if:
solutions of $(PDE)-(IC3)$ converge to the solutions of $(PDE)-(IC2)$ when $\sigma \rightarrow 0$, and
solutions of $(PDE)-(IC2)$ converge to the solutions of $(PDE)-(IC1)$ when $a \rightarrow \infty$?
For the problem I am currently working on it would be great if the answer to the question number 2. is yes. And if the answer to the question number 1. is yes I would use it in the future.
Some good reference in the literature would be very useful. And writing the answer would be great too. Thank you in advance.
