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

We study the homogeneous Dirichlet problem for the equation u(t) - div (F (z, & nabla;u) & nabla;u) = f, z= (x, t) is an element of Q(T) = omega x (0,T), where omega subset of R-N, is a bounded domain with & part;omega is an element of C-2, and F (z, xi) = a(z)|xi| (p(z)-2) + b(z)|xi|q((z)-2). The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that 2N/N+2 < p(z), q(z), and that the modulating coefficients and growth exponents satisfy the balance conditions a(z) + b(z) >= alpha > 0, |p(z) - q(z)| < 2/N + 2 in Q (over bar)(T) with alpha = const. We find conditions on the source f and the initial data u (middot,0) that guarantee the existence of a unique strong solution u with u(t) is an element of L-2 (QT) and a|& nabla;u|(p) + b|& nabla;u|(q) is an element of L-infinity (0, T; L-1 (omega)). The solution possesses the property of global higher integrability of the gradient, |& nabla;u|(min{p(z), q(z)} + r) is an element of L-1 (QT) with any r is an element of (0, 4/N+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: D-i (root F(z, & nabla;u) D(j)u) is an element of L-2 (QT), i=1, 2, ..., N. The same results are obtained for the equation with the regularized flux F(z, root epsilon(2)+(xi, xi)) xi