6.4.2: Uniqueness

Sufficiently regular solutions of the initial-boundary value problem (6.4.1)-(6.4.3) are uniquely determined since from

$$\begin{eqnarray*} c_t&=&D\triangle c\ \ \mbox{in}\ \Omega\times (0,\infty)\\ c(x,0)&=&0\\ \frac{\partial c}{\partial n}&=& 0\ \ \mbox{on}\ \partial\Omega\times (0,\infty). \end{eqnarray*}$$
it follows that for each $\tau>0$
$$\begin{eqnarray*} 0&=&\int_0^\tau\ \int_\Omega\ \left(c_tc-D(\triangle c)c\right)\ dxdt\\ &=&\int_\Omega\ \int_0^\tau\ \frac{1}{2}\frac{\partial}{\partial t}(c^2)\ dtdx+D\int_\Omega\ \int_0^\tau\ |\nabla_xc|^2\ dxdt\\ &=&\frac{1}{2}\int_\Omega\ c^2(x,\tau)\ dx+D\int_\Omega\ \int_0^\tau\ |\nabla_xc|^2\ dxdt. \end{eqnarray*}$$

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.