In a rod length L with insulated lateral surface and thermal constants $\displaystyle c, \ \rho, \ \kappa, \ K$ heat is generated at a rate uniformly proportional to the temperature, that is, at a rate $\displaystyle ru(x,t)$ per unit volume per unit time, where r is a constant and $\displaystyle u(x,t)$ is the temperature function of the rod. The ends of the rod are maintained at temperature 0 and initially the rod has uniform temperature 1.

Formulate an initial-BV problem for determining $\displaystyle u(x,t)$

$\displaystyle \text{D.E.}: \ u_t=ru_{xx}$

$\displaystyle \displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u( L,t)=0\end{cases}$

$\displaystyle \text{I.C.}: \ u(x,0)=1$

I am not sure what to do with the r so I just put it in front of u_{xx}. Is that correct? Also, I need a K, c, and rho somewhere but not sure where and why.