I would like to know what is a regularity of heat equation on a subdomain where the external force is more regular, for example, if the external force is zero in a subdoman. Let me clarify my question.
Let $T>0$, $\Omega\subset \mathbb{R}^n$ and $\omega\subset\subset \Omega$ be an open set. Let us denote by $Q=\Omega\times (0,T)$ and $\Sigma=\partial \Omega\times (0,T)$.
Consider the problem $$\begin{cases} y_t-\Delta y =v1_\omega,\ \text{ in } Q,\\ y\equiv 0,\ \text{ in } \Sigma,\\ y(x,0)=y_0(x), \ x\in \Omega, \end{cases}$$ where $y_0\in H^2(\Omega)$, $v\in L^2(\omega\times (0,T))$ and $1_\omega$ is the characteristic mapping, that is, it is zero outside of $\omega$ and 1 inside of $\omega$.
We know that this problem has a unique weak solution $y$ such that $$y\in C^0([0,T];L^2(\Omega))\cap L^2(0,T;H_0^1(\Omega)).$$
My question is: Since $v\equiv 0$ in $\Omega\setminus \omega$, given $\Omega'$ a open set, with smooth boundary, such that $\omega \subset\subset \Omega'$, then what is the regularity of $y$ on $\Omega\setminus \Omega'$? In other words, if we define $u=y\vert_{\Omega\setminus\Omega'}$, then $u$ is a weak solution to the equation $$u_t-\Delta u=0,\text{ in } (\Omega\setminus\Omega')\times (0,T).$$ Hence, does $u$ have better regularity? Can I conclude that $u\in L^\infty(0,T;\Omega\setminus\Omega')$ and that $$\operatorname{ess}\,\! \sup_{0\leq t\leq T}\|u(t)\|_{H^2(\Omega\setminus\Omega')}\leq C\|y_0\|_{H^2(\Omega\setminus\Omega')}?$$
Note that, From Evan's book Theorem 5, if $u$ is a weak solution to the problem $$\begin{cases} u_t-Lu =f,\ \text{ in } Q,\\ \color{blue}{ u\equiv 0,\ \text{ in } \Sigma,}\\ u(x,0)=g(x), \ x\in \Omega, \end{cases},$$ where $L$ is an elliptic operator, $u\in H^2(\Omega)$, $f\in H^1(0,T;L^2(\Omega))$ then $u\in L^\infty(0,T;\Omega)$ and that $$\operatorname{ess}\,\! \sup_{0\leq t\leq T}\|u(t)\|_{H^2(\Omega)}\leq C\left(\|f\|_{H^1(0,t;L^2(\Omega)}+\|g\|_{H^2(\Omega)}\right).$$
But here, we have that $u\equiv 0$ in $\Sigma$. When I consider $\Omega\setminus \Omega'$ we have $v\equiv 0$, however we have to consider other boundary condition $\partial \Omega'$. Is there a way to overcome this?