On the property of higher integrability for parabolic systems of variable order of nonlinearity
Vasilii Vasil'evich Zhikov, Svetlana Evgenievna Pastukhova · Mathematical Notes · 2010
We study a parabolic system of the form ∂ t u = div x A(x, t, ▿ x u) in a bounded cylinder Q T = Ω × (0, T) ⊂ ℝ +1 . Here the matrix function A(x, t, ζ) is subject to the conditions of power growth in the variable ζ and coercitivity with variable exponent p(x, t). It is assumed that p(x, 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 cylinderQ 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.