Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary.
Is this true:
Any solution $u(x,t)\in W^{2,p}$ of the equation can be written as $$u(x,t)=k(x,t)\star u^0(x)$$ where $k$ is a green function (depends on $\Omega$).
Best Answer
There exists a theory of Green functions for general parabolic boundary value problems which covers the case you are interested in, in particular papers by Eidelman, Ivasishen, Solonnikov. For references see
S. D. Eidelman and N. V. Zhitarashu, Parabolic boundary value problems. Basel: Birkhäuser (1998).
Unfortunately, most of the papers on this subject are available only in Russian.