Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Abstract

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation ut−∑i=1NDx(|Dxu|pDxu)=f(x,t) in the cylinder Ω×(0,T), where Ω⊂RN, N≥2, is a parallelepiped. The exponents of nonlinearity pi are given Lipschitz-continuous functions. It is shown that if [Formula presented], [Formula presented] then the problem has a unique solution u∈C([0,T];L2(Ω)) with |Dxu|p∈L∞(0,T;L1(Ω)), ut∈L2(QT). Moreover, [Formula presented] The assertions remain true for a smooth domain Ω if pi=2 on the lateral boundary of QT.