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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.

AxiomaticApproach
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  • If $u = u(t)$, aren't you just solving $$u' = \lambda u - u^{3}$$ as the Laplacian vanishes? – Matthew Cassell Dec 18 '18 at 03:12
  • @Mattos This was exactly what the question says, but I think in the context of the literature, the function does still depend on $x$, it just maps each $t$ to a function of $x$, otherwise the rest of the question would not really make any sense – AxiomaticApproach Dec 18 '18 at 20:48
  • I'm not sure what literature you're referring to, nor what you mean by 'the function does still depend on $x$, it just maps each $t$ to a function of $x$'. Think of the ODE as fixing a single point $\boldsymbol{x} = \boldsymbol{x_{0}} \in \mathbb{R}^{n}$ and watching the evolution of the PDE at that single point. – Matthew Cassell Dec 19 '18 at 10:04

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