I was told by a colleague that for some nonlinear PDEs the initial conditions can change the steady-state solution.
So can a stable steady solution depend on the initial conditions for nonlinear PDEs?
This goes against my understanding of differential equations. I suppose this is related to not being guaranteed uniqueness. I am doing numerical simulations where I have coupled multiple models together that are nonlinear.
I feel I have seen this effect in my model. This is deeply worrying to me as any predictive useful model has to be unique. Certainly, the transient should depend on initial conditions but I can't imagine a physical scenario where a stable steady-state solution could depend on initial conditions.
Yes, this can (and does) happen.
Consider the following (separable!) ODE: $$\frac{dU}{dt}=U-U^3\tag{1}$$ Solutions to (1) are one of the following (for some constant of integration $C$): \begin{gather*} U(t)=0 \\ U(t)=\pm(1+Ce^{-2t})^{-\frac{1}{2}} \end{gather*} In particular, as $t\to\infty$, we have $$U(t)\to\begin{cases} 1 & U(0)>0 \\ 0 & U(0)=0 \\ -1 & U(0)<0 \end{cases}$$
To adapt this to the PDE case, just consider (1) as describing the partial derivative of a function $U(x)$ with respect to time.