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:
$$f(x)= \delta_{x,L/2} $$