Solution of Heat equation with Neumann BC in an arbitrary domain

2012-02-21T03:38:13Z

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 $\Omega$).

Answer by Anatoly Kochubei for Solution of Heat equation with Neumann BC in an arbitrary domain

2012-02-21T15:28:06Z

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.

Solution of Heat equation with Neumann BC in an arbitrary domain - MathOverflow

Solution of Heat equation with Neumann BC in an arbitrary domain

Answer by Anatoly Kochubei for Solution of Heat equation with Neumann BC in an arbitrary domain