In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times \(t=r\in \mathbb {R}\) are often called PDE entire or eternal. For a nonlinear example, consider the quadratic parabolic PDE * \(\begin{aligned} w_t=w_{xx}+6w^2-\lambda , \end{aligned}\) for \(0<x<\tfrac{1}{2}\) , under Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits \(\Gamma (r)\) of (*) are heteroclinic among equilibria \(w=W_n(x)\) . For parameters \(\lambda >0\) , the trivial homogeneous equilibria are locally asymptotically stable \(W_0=-\sqrt{\lambda /6}\) , and \(W_\infty =+\sqrt{\lambda /6}\) of unstable dimension (Morse index) \(i(W_\infty )=1,2,3,\ldots \) , depending on \(\lambda \) . All nontrivial real \(W_n\) are rescaled and properly translated real-valued Weierstrass elliptic functions with Morse index \(i(W_n)=n\) . We show that the complex time extensions \(\Gamma (r+\textrm{i}s)\) , of analytic real heteroclinic orbits \(\Gamma (r)\) towards \(W_0\) , are not complex entire. For example, consider the time-reversible complex-valued solution \(\psi (s)=\Gamma (r_0-\textrm{i}s)\) of the nonlinear and nonconservative quadratic Schrödinger equation \(\begin{aligned} \textrm{i}\psi _s=\psi _{xx}+6\psi ^2-\lambda \end{aligned}\) with real initial condition \(\psi _0=\Gamma (r_0)\) . Then there exist real \(r_0\) such that \(\psi (s)\) blows up at some finite real times \(\pm s^*\ne 0\) . Abstractly, our results are formulated in the setting of analytic semigroups. They are based on Poincaré non-resonance of unstable eigenvalues at equilibria \(W_n\) , near pitchfork bifurcation. Technically, we have to except discrete sets of parameters \(\lambda \) , and are currently limited to unstable dimensions \(i(W_n)\le 22\) , or to fast unstable manifolds of dimensions \(d<1+\tfrac{1}{\sqrt{2}}i(W_n)\) .
