On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity

V. V. Zhikov ; S. E. Pastukhova

Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 179-200

Résumé

We study a parabolic system of the form $\partial_tu=\operatorname{div}_xA(x,t,\nabla_xu)$ in a bounded cylinder $Q_T=\Omega\times(0,T)\subset\mathbb R^{n+1}_{x,t}$. Here the matrix function $A(x,t,\xi)$ is subject to the conditions of power growth in the variable $\xi$ 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 $$ \frac{2n}{n+2}\alpha\le p(x,t)\le\beta\infty. $$ 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.

Détail

Keywords: parabolic system of variable order of nonlinearity, higher integrability for parabolic systems, Cacciopolli's inequality, Sobolev–Poincaré inequalities, Hölder's reverse inequality, Gehring's lemma, Sobolev–Orlicz space, Orlicz space.
Mots-clés : Lebesgue space