Heat equation with Neumann BC - MathOverflow

Heat equation with Neumann BC

2011-10-18T21:24:31Z

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.

Is this true to say:

$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ are two solutions of the heat equation in $W^{2,p}$.

Answer by Russell Brown for Heat equation with Neumann BC

2011-10-19T00:21:54Z

yes, with some regularity on the boundary.

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.

Answer by Rafa for Heat equation with Neumann BC

2013-05-15T10:35:48Z

A very naive answer:

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.