Existence of weak solutions of doubly nonlinear parabolic equations

We deal with a Cauchy-Dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space-time cylinder for doubly nonlinear parabolic equations, whose prototype is∂tu−div(|u|m−1|Du|p−2Du)=f with a non-negative Lebesgue function f on the right-hand side, where p>2nn+2 and m>0. The central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f. The constructed solution is obtained by a limit of approximations, i.e. we use solutions of regularized Cauchy-Dirichlet problems and pass to the limit to receive a solution for the original Cauchy-Dirichlet problem.