# Question on parabolic smoothing of nonhomogeneous heat equation

Suppose $$\Omega$$ is a bounded domain in $$\mathbb R^n$$ and let $$u$$ be the weak solution of the initial value problem for the nonhomogeneous heat equation: $$\begin{cases} \partial_t u-\Delta u=f,\;\;\;t\in (0,T)\\ u(x,0)=u_0\\ \end{cases}$$

If $$f\in L^1(\Omega \times (0,T))$$ then what assumption should we have on $$u_0$$ in order to deduce a bound for the solution $$u$$ in the parabolic Sobolev space?

To be honest, I'm not even sure if we can obtain any bound for the given $$f$$ but since we have one for the Poisson equation by Stampacchia(Proposition 4.3), I 'm motivated to think that it should exist also one for the heat equation (most of the times, parabolic pdes have analog results to the elliptic ones).

However I wasn't able to find anything, so this is why I'm asking here.

Any help or hint will be much appreciated.

• Most techniques are based on casting the equation to a funcional analysis settings involving inner products and dual spaces. You paper also requires $f \in L^2$, so it would be reasonable to assume the same. Otherwise, expressions like $<f,\phi>$, where <,> denotes the inner product, do not make sense. And any weak solutions are based on being multiplied by a test function $\phi$. Most bounds I know seem to involve a boundary condition term. Without one, it might be hard to bound all solutions uniformly. – F. Conrad Feb 28 at 1:01