Physical meaning of a heat equation with a term $\alpha u$

Question

I would like to know if there is any physical meaning for the equation $$ \begin{cases} u_t - u_{xx} + \alpha u = f(x), (a,b)\times(0, +\infty) \\ u(x,0)=u_0(x), x \in (a,b)\\ u(a,t)=u(b,t)=0, t \in (0,\infty) \end{cases} . $$ When $\alpha=0$ we obtain the classical heat equation and the meaning is very well. However when $\alpha \neq 0$ I don't know if the equation makes sense in phisical point of view.

Answer 1

So the answer is two-fold.

First, mathematicians like to f*** around with differental equations and just add/tweak/generalize/... stuff just to see if they can still prove existence and other properties.

Second, this equations actually has some interesting modelling applications but it is no longer suitable to model temperature. Basically you have the $\Delta u$-term which describes diffusion and the $\alpha u$-term which describes exponential growth. Those two effects usually work against each other so you get interesting solutions. A more general class of equations $u_t=\Delta u + f(u)$is usually called Reaction-Diffusion-euqations.