On the property of higher integrability for parabolic systems of variable order of nonlinearity (Q1957034)

The authors study a parabolic system of the form \(\partial _t u = \text{div}_x A(x, t, \nabla_x u)\) in a bounded cylinder \(Q_T = \Omega \times (0, T) \subset \mathbb R_{x,t}^{n+1}\). The matrix function \(A(x,t, \zeta)\) supposed to be Carathéodory and to be a subject to suitable power growth conditions in the variable \(\zeta \) and coercitivity with variable exponent \(p(x,t)\). It is assumed that \(p(x,t)\in C(Q_T)\) has a logarithmic modulus of continuity and satisfies the estimate \[ \frac{2n}{n + 2} < \alpha \leqslant p(x,t) \leqslant \beta < \infty. \] For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinder \(Q_T\). The method of a solution is based on a localization of a special kind and a local variant (adapted for parabolic problems) of Gehring's lemma with variable exponent of integrability proved in the paper.