I am trying to find a solution to the nonlinear reaction-diffusion PDE:
$$u_t = \Delta u + \lambda u - u^3$$
where $(x,t) \in \Omega \times (0,\infty)$ and $\Omega\subset \mathbb{R}^n$ is a bounded domain with smooth boundary. Additionally, the boundary condition $u = 0$ on $\partial \Omega$ should hold and $\lambda< 0$.
Apparently, one can find an ODE for a solution $u = u(t)$, but after a while of searching around, I could not find any. At first I thought about using the fourier transform, but the nonlinearity of the equation ruins this approach, as well as many other approaches I have learned so far. Integrating with respect to $x$ was another idea, but the boundary conditions won't let me partially integrate to make the laplacian vanish.
Any hints on how to find said ODE or how to tackle a problem like this generally would be kindly appreciated.