A scalar parabolic PDE f(x,u,u_x,u_xx,u_t)=0 on an interval can be shown to be gradient-like by building an energy E = integral of L(x,u,u_x) that strictly decreases away from equilibria. The construction solves the equation for the diffusion term u_xx by the implicit function theorem, splits it into an equilibrium part F0 and a strictly positive time-dependent factor F1, and then obtains the Lagrangian from L_pp = exp(g) where g solves a first-order PDE along characteristics; integration constants are chosen so the boundary terms vanish under Dirichlet or nonlinear Robin conditions. The payoff is that LaSalle's invariance principle applies, so bounded trajectories converge to equilibria and the global attractor decomposes into equilibria and heteroclinic orbits.
This paper by Phillipo Lappicy and Bernold Fiedler constructs explicit Lyapunov functions for fully nonlinear parabolic equations in one spatial dimension, extending classical techniques from Zelenya…