Directedness of solution set for some quasilinear multi-valued parabolic problems

Recently the authors proved the existence of solutions within a sector of appropriately defined sub-supersolution for a general class of quasilinear parabolic inclusions with Clarke's generalized gradient. The main goal of this article is to show that the solution set contained in this sector is a directed set with respect to the natural partial ordering of functions. Until now the directedness was obtained by assuming a certain additional one-sided growth condition on Clarke's generalized gradient, and it was still an open problem as to whether one can completely drop this condition. Here we provide an affirmative answer, saying that one can indeed drop this additional condition, and only a local Lq-boundedness condition on Clarke's gradient is sufficient. Unlike in the corresponding elliptic case, we are faced with the additional difficulty that the underlying solution space does not possess lattice structure, which considerably complicates the proof of directedness in the parabolic case considered here.