Solving $u_t-u_{xx}=u(1-u)$ with initial/boundary conditions How would one go about solving, or describing the solutions to this non-linear PDE, the heat equation with an extra non-linear term.
$$u_t-u_{xx}=u(1-u)$$
Suppose,
$$u=u(x,t),\quad x\in[0,L],\quad u(L,t)=0=u(0,t), \quad u(x,0)=f(x)$$
However i'm not too worried about specific initial/boundary conditions, just looking for a nice way to attack this.
For instance perhaps:
$$f(x)= \delta_{x,L/2} $$
 A: 
just looking for a nice way to attack this.

Would you like a pony with that? It's a nonlinear PDE. 

How would one go about solving, or describing the solutions

By reading the existing literature on Fisher's equation (which is what it is, as pointed out in comments). It is known to have traveling wave solutions for various speeds, and — miraculously — solutions with speed $5/\sqrt{6}$ have an explicit form such as 
$$u(x,t) = \left\{1+\exp\left(\frac{x}{\sqrt{6}} -\frac{5t}{6}\right) \right\}^{-2} $$ This solution is pictured below (times $t=-5,-4,\dots,4,5$ are represented by colors changing  from magenta to cyan): 

Looks like a tide that never goes back down. 
For more, see EqWorld entry and the references cited there, in particular 


*

*Danilov, V. G., Maslov V. P., and Volosov, K. A., Mathematical Modelling of Heat and Mass Transfer Processes, Kluwer, Dordrecht, 1995.

*Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, 2004.

