Existence and Uniqueness for Doubly Nonlinear Parabolic Equations with Nonstandard Growth Conditions

We study the homogeneous Dirichlet problem for the equation ut = n ∑ i=1 Di ( ai|Di(|u|m(x)−1u)|pi(x,t)−2Di(|u|m(x)−1u) ) +b|u|σ(x,t)−2u with given exponents m(x) , pi(x,t) and σ(x,t) . It is proved that the problem has a solution in a suitable variable exponent Sobolev space. In dependence on the properties of the coefficient b and the exponents of nonlinearity, the solution exists globally or locally in time. The comparison principle and uniqueness are proved under additional restrictions on the data.