Heat equation with Neumann BC - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:46:43Z http://mathoverflow.net/feeds/question/78488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78488/heat-equation-with-neumann-bc Heat equation with Neumann BC unknown (google) 2011-10-18T21:24:31Z 2013-05-15T10:35:48Z <p>Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.</p> <p>Is this true to say:</p> <p>$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ are two solutions of the heat equation in $W^{2,p}$.</p> http://mathoverflow.net/questions/78488/heat-equation-with-neumann-bc/78514#78514 Answer by Russell Brown for Heat equation with Neumann BC Russell Brown 2011-10-19T00:21:54Z 2011-10-19T00:21:54Z <p>yes, with some regularity on the boundary. </p> <p>Theorem 3.2.9 p. 90 of E.B. Davies book, Heat Kernels and Spectral Theory gives Gaussian bounds for the heat kernel of an elliptic operator with Neumann boundary conditions. These bounds imply that the heat flow preserves L^p. </p> http://mathoverflow.net/questions/78488/heat-equation-with-neumann-bc/130691#130691 Answer by Rafa for Heat equation with Neumann BC Rafa 2013-05-15T10:35:48Z 2013-05-15T10:35:48Z <p>A very naive answer:</p> <p>Assume that the initial data is positive (the same should be true dealing with absolute values...) and take p>2 (p=2 follows the same idea). Multiply the equation by $pu^{p-1}$ and integrate. One gets $$p\int_\Omega u^{p-1}u_t dx=\frac{d}{dt}\int_\Omega u^p dx=p\int_\Omega \Delta u u^{p-1}dx=-p\int_\Omega \nabla u\cdot((p-1)u^{p-2}\nabla u)\leq0,$$ where in the last equality we use Green formula an homogeneous Neumann BC. Due to the linearity of the equation one gets the same result for the difference of two solutions.</p>