Abstract:
We study a parabolic system of the form ∂tu=divxA(x,t,∇xu) in a bounded cylinder QT=Ω×(0,T)⊂Rn+1x,t. Here the matrix function A(x,t,ξ) is subject to the conditions of power growth in the variable ξ 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
2nn+2<α⩽p(x,t)⩽β<∞.
For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinder QT. 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.
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, Lebesgue space, Sobolev–Orlicz space, Orlicz space.
Citation:
V. V. Zhikov, S. E. Pastukhova, “On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity”, Mat. Zametki, 87:2 (2010), 179–200; Math. Notes, 87:2 (2010), 169–188
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Linking options:
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This publication is cited in the following 28 articles:
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