Fix $T>0$. Let us consider a heat equation $\rho(t,x)$ with initial condition $\rho_0(\cdot):[0,1]\to\mathbb R$ on the interval $[0,1]$. Namely, $$\partial_t\rho(t,x)=\Delta \rho(t,x)$$ for every $x\in(0,1)$, $t\in(0,T]$, and $$\rho(0,x)=\rho_0(x)$$ for every $x\in(0,1)$.
Fix $a,b>0$ and assume that $\rho_0(0)=a$ and $\rho_0(1)=b$. Let us impose three types of boundary conditions:
- Dirichlet type: $\rho(t,0)=a, \, \rho(t,1)=b, $ for every $t\in[0,T]$
- Robin type: $\partial_x^+\rho(t,0)= \rho(t,0),\partial_x^-\rho(t,1)= -\rho(t,1),$ for every $t\in[0,T]$
- Neumann type: $\partial_x^+\rho(t,0)=\partial_x^-\rho(t,1)= 0$ for every $t\in[0,T]$
My question is very simple(maybe silly): For all these three types of boundary conditions, is the classical solution $\rho(t,\cdot)$ is smooth on $[0,1]$, if the initail condition $\rho_0$ is smooth on $[0,1]$? Here a function $f$ is smooth on $[0,1]$ means that for every $m\geq 0$, the $m-th$ derivative $f^{(m)}$ is continuous on $[0,1]$, where we take $\partial_x^+f(0)$ as $\partial_xf(0)$ and $\partial_x^-f(1)$ as $\partial_x f(1)$.
It seems to be a very natural question, but I could not find any reference discussing about it. What I have tried is as follows: we could write down the solution by the method of separation of variables, as a infinite series. But I could not show that the space derivative of $\rho(t,\cdot)$ is equal to the infinite sum of the space derivative of each term, because uniform convergence does not hold.
Any idea or reference is much appreciated.