In a rod length L with insulated lateral surface and thermal constants heat is generated at a rate uniformly proportional to the temperature, that is, at a rate per unit volume per unit time, where r is a constant and 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

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.