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Mathematics LibreTexts

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*}

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