Solution of Heat equation with Neumann BC in an arbitrary domain - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:40:26Z http://mathoverflow.net/feeds/question/89078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89078/solution-of-heat-equation-with-neumann-bc-in-an-arbitrary-domain Solution of Heat equation with Neumann BC in an arbitrary domain unknown (google) 2012-02-21T03:38:13Z 2012-03-20T17:22:01Z <p>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:</p> <p>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 $\Omega$). </p> http://mathoverflow.net/questions/89078/solution-of-heat-equation-with-neumann-bc-in-an-arbitrary-domain/89114#89114 Answer by Anatoly Kochubei for Solution of Heat equation with Neumann BC in an arbitrary domain Anatoly Kochubei 2012-02-21T15:28:06Z 2012-02-21T15:28:06Z <p>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 </p> <p>S. D. Eidelman and N. V. Zhitarashu, Parabolic boundary value problems. Basel: Birkhäuser (1998).</p> <p>Unfortunately, most of the papers on this subject are available only in Russian.</p>