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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.

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1 Answer 1

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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.

I am not too sure how helpful this answer is to you since looking at the proofs in these reference is not a very fun thing to do. Especially the second one is so general that is (in my opinion) quite hard to read. However, they are my go-to references for such questions.

Perhaps a better reference is Superlinear parabolic problems by Quittner and Souplet. They handle your equation in detail and somewhere (in the appendix, maybe) prove certain regularity results.

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