# Regularity of Parabolic pde (via Boostrap argument?) and references needed

Let $$M$$ be a compact $$2d$$-manifold in $$\mathbb R^3$$ and take $$T>0\;$$. We set $$M_T:=M \times (0,T)$$ and we consider the following parabolic PDE

\begin{align} \partial t u-\Delta_M u&=f \quad \text{ in } M_T \tag 1 \\ u(\cdot,0)&=u_0 \quad \text{in } M \tag 2 \end{align}

Assumptions:

• The right-hand side of $$(1)$$ is a product function of the form $$f=g\times h \in L^\infty(M_T)\times C^2(M)$$
• The initial data can be as regular and smooth we wish, for instance $$C^4(M)$$

EDIT: $$h$$ cannot be constant. It is a function depending only on space variable $$x$$.

Goal: To find how regular and smooth $$u$$ is.

Questions:

1. What is the exact regularity of $$f$$? In order to estimate the regularity of $$u$$, I should first clarify this part. However although I know I should expect something "good" for $$f$$, I don't know how "good" exactly $$f$$ can be. Is this product function $$C^2$$? Is there any helpful "rule" in order to recognize "easily" where a product function lies?
2. I am aware of theorems that provide $$W^{2,1}_p$$-estimates for solutions of problem $$(1),(2)$$ as long as $$f \in L^p$$ and $$u_0\in W^{2,p}$$ but I don't know any theorems or references where the right-hand side has better regularity than $$L^p$$. Is there any reference for that or it follows by a bootstrap argument? For example, if $$f$$ was $$C^1(M_T)$$, then is there any reference/theorem proving that $$u$$ is $$C^3$$ in space and $$C^1$$ in time? Or would this follow by a bootstrap argument? How would this argument work then?

I hope I made my questions clear enough because I always have a hard time getting my head around regularity issues. I would appreciate if someone could answer these questions and I apologize in advance if they sound too silly\elementary.

Many many thanks for the time!

• There is no way $f$ can have better regularity than $L^\infty$: there is no conditions on $h$. Nov 18, 2021 at 12:35
• @ArcticChar What do you mean there is no condition on $h$? You mean for time? Nov 18, 2021 at 12:38
• Let say $h=1$. Then $f=g$ is in $L^\infty$ and not in $C^1$. Nov 18, 2021 at 12:44
• @ArcticChar well $h$ can not be constant. I will add it in the post Nov 18, 2021 at 12:47
• Does it matter? Whenever $h$ is non-zero in a neigborhood, $g = f/h$. So if $g$ is not continuous, $f$ cannot be continuous there too. Nov 18, 2021 at 12:55

Bootstrapping. The bootstrapping method applies only if $$f$$ depends on $$u$$. More precisely, if improved regularity of $$u$$ implies improved regularity of $$f$$. The way you described your problem, there's no hope for this, so I wouldn't expect a bootstrapping argument here. Your boots don't have straps for us to pull.
Regularity of $$f$$. The way it is now, $$f$$ is clearly $$L^\infty$$ (and nothing more).
Regularity of $$u$$. You already noted that $$u \in W^{2,p}$$ for any $$p < \infty$$. By Morrey's embedding, also $$u \in C^{1,\alpha}$$ for any $$\alpha < 1$$. And I wouldn't expect more.
It would be tempting to think that $$f \in L^\infty \Longrightarrow \nabla^2 u \in L^\infty \quad \text{and/or} \quad f \in C^0 \Longrightarrow u \in C^2.$$ But these implications are false. In fact (see Gilbarg and Trudinger's book, problem 4.9) there's $$f \in C^0$$ such that any solution $$v$$ of $$-\Delta v = f$$ has unbounded second derivatives. If you take $$u(t,x) = v(t,x)$$, you obtain a solution $$u \notin C^2$$ of $$(\partial_t-\Delta) u = f$$.
• First of all, thanks a lot for the answer. Since I am not familiar with this "bootstrapping" technique, I might erroneous ommited that indeed in my case $f$ depends on $u$. So then, since $u$ is uniformly continuous (due to the Hölder regularity), could we probably say more about the regularity of $f$ or at least have these straps for pulling as you suggested? Nov 18, 2021 at 14:30
• Well, if $f$ in fact is a function of $u$, that's a whole new question. I'm not an expert, but I think the answer depends on the details. Nov 18, 2021 at 14:34