Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $\alpha$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty} u(x,t)=0$$
This studying case requires to demonstrate that isoprobability contours, it means, in the points (x,t) which P(x,t)=P=constant is verified that $$\frac{x}{t}=\pm [4\alpha k-2k\frac{\log(t)}{t}-\frac{4k}{t}\log(\sqrt{4\pi k} P)]^\frac{1}{2}$$
Another aspect to demonstrate is that $t \to \infty$, the spreading velocity of these contours, it means, the velocity which these organisms are spreading is aproximated to $$\frac{x}{t}=\pm(4\alpha k)^\frac{1}{2}$$
Finally, how to compare this spreading velocity with purely diffusive process $(\alpha=0)$, it means , x is aproximated to $\sqrt{kt}$
This is just for academical curiosity and I would like to understand better this kind of cases with Partial Differential Equations. So, I require any guidance or starting steps or explanations to find the solutions because it's an interesting problem.
Thanks very much for your attention.
Multiply both sides of your equation by $e^{-\alpha t}$. Then you get $$\frac{\partial u}{\partial t}e^{-\alpha t}-\alpha u\, e^{-\alpha t}=\frac{\partial^2 u}{\partial x^2}e^{-\alpha t}.$$ We recognize the product rule on the left-hand side: $$\frac{\partial}{\partial t}(ue^{-\alpha t})=\frac{\partial^2}{\partial x^2}(ue^{-\alpha t}).$$ So if $v$ solves the diffusion equation (with the same boundary conditions), then the function you want is $u=ve^{\alpha t}.$ Therefore, to solve your problem, you only need to solve the diffusion equation, whose solutions are well-understood.