Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

Abstract

The aim of this paper is to establish a Meyer's type higher integrability result for weak solutions of possibly degenerate parabolic systems of the typeThe vector-field a is assumed to fulfill a non-standard p(x, t)-growth condition. In particular it is shown that there exists ε > 0 depending only on the structural data such that there holds:loc , together with a local estimate for the p(·)(1 + ε)-energy.

Very weak solutions of subquadratic parabolic systems with non-standardp(x,t)-growth

“…The argument uses a certain stopping time argument which allows to construct a covering of the upper level sets. This method has its origin in [12,13]; a slightly simplified version can be found in [5] and [7]. Since most of the arguments are standard by now, 28 Q. LI we will only give the main ideas to the proof and refer to [7, section 9] and [5, §7] for the details.…”

“…Later on Zhikov and Pastukhova [15] and independently Bögelein and Duzaar [5] proved the higher integrability of weak solutions to parabolic systems with nonstandard p(x, t)-growth whose model is the parabolic p(x, t)-Laplacian system:…”

“…[9]. Our proof is in the spirit of [7,13] and we will work with a "non-standard version" of the intrinsic geometry introduced by Bögelein and Duzaar [5]. However, the proof in [7] strongly depended on the assumption that p(x, t) ≥ 2.…”

“…However, the proof in [7] strongly depended on the assumption that p(x, t) ≥ 2. The major difficulty in our subquadratic case is that the parabolic cylinder studied in [5,7] cannot directly apply to the subquadratic case, since Q instead to deal with our subquadratic case. We also remark that as pointed out by Kopaliani [14], the strong maximal functions are not bounded in L p (·) unless p(·) ≡ constant.…”

Regularity results for nonlinear elliptic systems

“…The parabolic analogue of the result from Chapter 6 appeared in [DH12]. This latter work was made possible by higher integrability estimates found in [BD11], which was developed independently from [ZP10], building upon the techniques of [KL00] to provide the parabolic analogue to [Zhi97].…”