Solving the reaction-diffusion equation for a single species

Question

$$ \frac{\partial u}{\partial t} =k\Delta u+ru. $$ Where all of the bounds are $0$.

Please help! Very new to PDE's and don't understand how to solve this. I know that I need to use separation of variables somehow.

Answer 1

You can change variables to arrive to the standard heat equation. Consider $$ v(x,t)=e^{-rt}u(x,t). $$ If you multiply the original equation by $e^{-rt}$, you get $$ \partial_t v=\partial_t u e^{-rt}-re^{-rt}u=e^{-rt}\Delta u=\Delta v $$ which is the classical heat equation. Now you can solve by separation of variables easily.