The connection problem for fully nonlinear parabolic equations in one spatial dimension

The paper’s Theorem 1.1 explicitly asserts that for fully nonlinear parabolic PDEs f(x,u,ux,uxx,ut)=0 with Neumann boundary conditions, under parabolicity fq·fr<0, a bounded dissipative precompact semiflow, and hyperbolic equilibria, the global attractor A is the union of finitely many equilibria and heteroclinic orbits; moreover, a heteroclinic u−→u+ exists if and only if u− and u+ are adjacent and i(u−)>i(u+) (see Theorem 1.1 and surrounding discussion ). The gradient structure via a Lyapunov function ensures ω-limit sets are single equilibria, yielding A=E∪H (equilibria and heteroclinics) . The zero-number (lap-number) monotonicity is established for fully nonlinear flows by linearizing the difference of two solutions to a(t,x)vxx + b(t,x)vx + c(t,x)v with a(t,x)>0 and bounded coefficients (equations (2.6)–(2.7)), giving the standard drop at multiple zeros . The paper also states the blocking and liberalism principles (with a fully nonlinear Conley-index homotopy) and Wolfrum’s equivalence to bridge adjacency notions, completing the if-and-only-if criterion . The candidate solution mirrors these steps: implicit-function reductions to quasilinear forms, zero-number monotonicity, gradient/ω-limit consequences, index drop, boundary-based blocking, and the liberalism/adjacency theorem. Any small differences are stylistic (e.g., the model sketches a degree/continuation argument, while the paper provides an explicit Conley-index homotopy).