How to simplify the heat equation $u_{t}-\Delta u=\sin(u)$by substitution

Question

Given $u_{t}-\vartriangle u=\sin(u)$ for $x \in \Omega$ and $t>0$, how would one substitute $u$ with a function $v$ such that this equation reduces to the form $v_{t}-\vartriangle v=0$?

Answer 1

I don't believe that this is possible. The reason is that for suitable $\Omega$ (e.g. large intervals) the stationary equation $-\Delta u = \sin x$ has more than solution under zero Dirichlet boundary conditions while the equation $-\Delta v = 0$ has only one solution.

More loosely arguing, if this were possible, there should also be a similar transformation that turns solutions of the scalar ode $u' + \lambda u = \sin u$ into solutions of $v' + \lambda v = 0$, for arbitrary positive $\lambda$. But the zero solution of the first equation is unstable for $\lambda < 1$ while it is always stable for the second one.