Uniqueness of weak solutions

Establish uniqueness of weak solutions to the two-dimensional Davis equation with degenerate fourth-order flux under the stated no-flux boundary conditions and the regularity assumptions of Theorem 1, where the principal term is \(\nabla\cdot(h^3\nabla\Delta h)\).

Background

The paper proves global existence, non-negativity, and strict positivity almost everywhere for weak solutions of the two-dimensional Davis equation modeling surface-tension-driven convection in thin films. The weak-solution regularity obtained in Theorem 1 is $$h\in L^\infty(0,T;H^1(\Omega))\cap L^2(0,T;H^2(\Omega))$$, together with corresponding time-derivative and nonlinear-flux bounds.

The authors state that this regularity is insufficient for standard $L^2$-contraction or energy-difference estimates involving the degenerate fourth-order term. Consequently, whether two weak solutions with the same initial data must coincide remains unresolved.