\(W^{1,p}\)-regularity of almost periodic solutions of nonlinear parabolic equations (Q1363267)

It is well-known that gradients of generalized solutions of the Dirichlet problem for nonlinear elliptic equations have heightened summability (estimates of Meyers type). In [\textit{K. Gröger}, Math. Ann. 283, No. 4, 679-687 (1989; Zbl 0646.35024)], a new approach to results of such type was proposed, which allows one to include domains with Lipschitz boundary and mixed boundary conditions (in the case of second-order equations). Results on the \(W^{1,p}\)-regularity of mixed problems for parabolic equations were obtained in [\textit{K. Gröger}, Nonlinear Anal., Theory Methods Appl. 18, No. 6, 569-577 (1992; Zbl 0764.35022)]. In this paper, a similar approach is used to investigate the \(W^{1,p}\)-regularity of bounded and almost periodic (a.p.) solutions of mixed problems for nonlinear parabolic second-order equations. We consider boundedness and Stepanov almost periodicity, Besicovitch almost periodicity, as well as, closely connected with the latter, the case of solutions that are summable with some degree on the entire axis. Note that in Stepanov's case, an additional quantitative condition on the operator arises that guarantees \(W^{1,p}\)-regularity. However, it remains vague whether or not it is necessary.