Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents

The first boundary value problem is considered for a certain class of anisotropic parabolic equations with variable nonlinearity exponents in a cylindrical domain (0, T) x Omega, where Omega is a bounded domain. The parabolic term in the equation has the form (beta(x,u))(t) and is determined by the function beta(x, r) is an element of L-1 (Omega), where r is an element of R, which only satisfies the Caratheodory condition and is increasing in r. The existence of a weak and a renormalized solution is proved. Bibliography: 26 titles.