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

We consider the homogeneous Dirichlet problem for anisotropic parabolic equationut−∑i=1NDxi(|Dxiu|pi(x,t)−2Dxiu)=f(x,t) in cylinder Ω×(0,T), where Ω⊂RN, N≥2, is a parallelepiped. The exponents of nonlinearity pi are given Lipschitz-continuous functions. It shown that if pi(x,t)>2NN+2,μ=supQT⁡maxi⁡pi(x,t)mini⁡pi(x,t)<1+1N,|Dxiu0|max⁡{pi(⋅,0),2}∈L1(Ω),f∈L2(0,T;W01,2(Ω)), then has unique solution u∈C([0,T];L2(Ω)) with |Dxiu|pi∈L∞(0,T;L1(Ω)), ut∈L2(QT). Moreover,|Dxiu|pi+r∈L1(QT)with some r=r(μ,N)>0,|Dxiu|pi−22Dxiu∈W1,2(QT). assertions remain true smooth domain Ω pi=2 on lateral boundary QT.