Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Abstract

We study the homogeneous Dirichlet problem for the equation ut-div(F(z,∇u)∇u)=f,z=(x,t)∈QT=Ω×(0,T),where Ω ⊂ RN, is a bounded domain with ∂Ω ∈ C2, and F(z, ξ) = a(z) | ξ| p(z)-2+ b(z) | ξ| q(z)-2. The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that 2NN+2<p(z),q(z), and that the modulating coefficients and growth exponents satisfy the balance conditions a(z)+b(z)≥α>0,|p(z)-q(z)|<2N+2inQ¯Twith α= const. We find conditions on the source f and the initial data u(· , 0) that guarantee the existence of a unique strong solution u with ut∈ L2(QT) and a| ∇ u| p+ b| ∇ u| q∈ L∞(0 , T; L1(Ω)). The solution possesses the property of global higher integrability of the gradient, |∇u|min{p(z),q(z)}+r∈L1(QT)with anyr∈(0,4N+2),which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven: Di(F(z,∇u)Dju)∈L2(QT),i=1,2,…,N.The same results are obtained for the equation with the regularized flux F(z,ϵ2+(ξ,ξ))ξ. © 2022, The Author(s).