  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
example, consider the quadratic parabolic PDE \begin{equation*} \label{*}
w_t=w_{xx}+6w^2-\lambda, \tag{*} \end{equation*} for $0<x<\tfrac{1}{2}$, under
Neumann boundary conditions. By its gradient-like structure, all real eternal
non-equilibrium orbits $\Gamma(r)$ of \eqref{*} are heteroclinic among
equilibria $w=W_n(x)$. All nontrivial real $W_n$ are rescaled and properly
translated real-valued Weierstrass elliptic functions $W_n(x)=-n^2
\wp(nx+\tfrac{1}{2}\tau),\ n=1,2,3,\ldots$, with lattice periods 1 and the
purely imaginary modular parameter $\tau$, and with Morse index $i(W_n)=n$ at
$\lambda=\tfrac{1}{2}n^4 g_2(\tau)$.
  We show that the complex time extensions $\Gamma(r+\mathrm{i}s)$, of the real
analytic heteroclinic orbits towards $W_0=-\sqrt{\lambda/6}$, are not complex
entire. For any fixed real $r_0$, for example, consider the time-reversible
complex-valued solution $\psi(s)$ of the nonlinear and nonconservative
quadratic Schr\"odinger equation \begin{equation*} \label{**}
\mathrm{i}\psi_s=\psi_{xx}+6\psi^2-\lambda \tag{**} \end{equation*} with real
initial condition $\psi_0=\Gamma(r_0)$. Then there exist $r_0$ such that
$\psi(s)$ blows up at some finite real times $\pm s^*$.
  Abstractly, our results are formulated in the setting of analytic semigroups.
They are based on Poincar\'e non-resonance of unstable eigenvalues at
equilibria $W_n$, near pitchfork bifurcation. Technically, we have to except a
discrete set of $\lambda>0$, and are currently limited to unstable dimensions
$n\leq22$, or to fast unstable manifolds of dimensions
$d<1+\tfrac{1}{\sqrt{2}}n$.
