Schauder regularity heat equation

Question

Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.

It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$, $g \in C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times\partial\Omega)$, $u_0 \in C^{m,\alpha}(\mathrm{cl}\, \Omega)$ (satisfying some compatibility conditions at $t=0$), then there exists a unique solution $u$ in $C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$ of \begin{cases} \partial_t u -\Delta u = f &\mbox{ in }[0,T]\times \mathrm{cl}\,\Omega,\\ u=g & \mbox{ on } [0,T] \times \partial\Omega,\\ u(0,\cdot) = u_0 & \mbox{ in }\mathrm{cl} \, \Omega. \end{cases}

For the elliptic case holds a very similar result, but in this case we allow $m$ to be also $1$, that is $m \in \mathbb{N}\setminus \{0\}$. If $f \in C^{m-2,\alpha}(\mathrm{cl}\,\Omega)$, $g \in C^{m,\alpha}(\partial\Omega)$, then there exists a unique solution $u$ in $C^{m,\alpha}(\mathrm{cl}\,\Omega)$ of \begin{cases} \Delta u = f &\mbox{ in }\mathrm{cl}\,\Omega,\\ u=g & \mbox{ on } \partial\Omega. \end{cases} In this case, for $m=1$, the space $C^{-1,\alpha}(\mathrm{cl}\,\Omega)$ is the space of distributions which euqals the divergence of an element in $C^{0,\alpha}(\mathrm{cl}\,\Omega,\mathbb{C}^n)$, and the Laplacian is to be intended in the weak sense.

Then my question is the following:

There exists an analog of the case $m=1$ for the heat (parabolic) equation?

Answer 1

For a partial answer, see Theorem 6.48 in Second order parabolic differential equations, 1996, by Gary Lieberman.

There is an additional assumption that $f$ belongs to the Morrey space $M^{1,n+1+\alpha}$ defined page 130 of the book. In particular, $L^{\infty}\subset M^{1,n+1+\alpha}\subset L^1$.

I do not know if you can remove this additional assumption or not.

Answer 2

A partial answer.

There are results for parabolic Schauder theory in $C^{1,\alpha}(\bar \Omega)$. Namely, for the rhs $f\equiv0$, $u_0\in C^{1,\alpha}$, $g\in C^{1,\alpha}$ see Baderko E.A. "Parabolic problems and boundary integral equations", Math. Methods Appl. Sci. 1997, V.20, P. 449-459. The results are for unbounded cylinder $\Omega\times[0,\infty)$ and for parabolic equations with Holder coefficients. There are references here for her earlier works in bounded wrt $t$ cylinders.

As for the rhs $f$, it is shown in M. F. Cherepova, “On some properties of the parabolic potential of bulk masses. I”, Differ. Uravn., 35:12 (1999), 1701–1706; Differ. Equ., 35:12 (1999), 1726–1732, if it satisfies the condition $|f|\le C d^{\alpha-1}$, where $d$ is the parabolic distance to the parabolic boundary, then the volume potential $Vf$ belongs to $C^{1,\alpha}$ in the closure of the domain and therefore solutions of the first BVP with such $f$ (and $u_0$, $g$) belong to the same class.