# $L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $$\mathbb{R}^n$$ with homogeneous Neumann BCs:

$$u_t - \Delta u = f$$ $$\partial_\nu u = 0$$ $$u|_{t=0} = u_0$$ where $$f \in L^p(0,T;L^r(\Omega))$$ and $$u_0 \in L^q(\Omega)$$ can be smoother if required.

Under which conditions on $$p,r,q$$ and $$n$$ do we get that the weak solution is in $$L^\infty((0,T)\times \Omega)$$?

I can find something in the LSU book -- but I'm sure there have been improvements. What's the maximal regularity result for this?

• I guess LSU is Ladyzhenskaya-Solonnikov-Uraltseva instead of google's suggestion (Louisiana State U) Sep 2, 2021 at 8:40
• You need $u_0 \in L^\infty$, since $u(t,⋅)→u_0$. If $Ω=R^n$, using the fundamental solution one has that u is bounded whenever $(n/2)(1−1/r′)p′<1$ and the same holds in the half space (using an even reflection one goes back to the whole space). Probably one cannot do better and the same should hold in regular domains, using Gaussian estimates. Maybe I am adding nothing to what you already know. Sep 2, 2021 at 18:07