# Parabolic Bootstrapping

I started studying parabolic pdes. Often I come across an integral solution where the regularity is proven by a standard bootstrap argument or by standard parabolic results, but it is never explained how this works in detail.

For example, let $$\Omega$$ be an open, bounded subset of $$\mathbb{R}^n$$ with smooth boundary. Let $$n>2$$, $$p \in (1, (n+2)/(n-2))$$, and $$T>0$$. Consider the nonlinear heat equation with initial data $$u_0 \geq 0$$

\begin{aligned} u_t - \Delta u &= u^p & &\text{in }\Omega\times(0,T), \\ u &= 0 & &\text{on } \partial\Omega\times(0,T), \\ u(x,0) &= u_0(x) & &\text{for all } x \in \Omega.\end{aligned}

Now I have an integral solution $$U$$ in the sense that $$U(x,t):\Omega\times(0,T) \rightarrow [0,\infty]$$ is a nonnegative measurable function such that

$$U(x,t) = \int_\Omega G(t,x,y) u_0(y) dy + \int_0^t \int_\Omega G(t-s,x,y) U^p(y,s) dy ds,$$

where $$G(t-s,x,y)$$ denotes the Green function of the heat equation with Dirichlet boundary condition. I also know that $$U \in L^\infty_{loc}((0,T);L^\infty(\Omega))$$. Now it says that one can show that $$U \in C^{2,1}(\bar{\Omega}\times(0,T))$$ with a standard bootstrap argument. Could someone explain how this works?

I have seen bootstrapping for elliptic pdes in the sense that we get higher regularity for a weak solution by applying Sobolev embeddings/schauder estimates over and over again. But it don't see how this would work in this case, because we have an integral solution and not a weak solution.

This follows from linear theory. Set $$f(x, t) = U^p(x, t)$$.
Bounded weak solutions are Hölder continuous (see vor instance Porzio–Vespri), hence $$f$$ is. Then one can apply Theorem IV 5.2 (or 5.1?) of Linear and quasi-linear equations of parabolic type by Ladyženskaja, Solonnikov and Ural'ceva to obtain the desired result.