Existence theorems for solutions of parabolic equations with variable order of nonlinearity

We study the solvability of an initial-boundary value problem for second-order parabolic equations with variable order of nonlinearity. In the model case, the equation contains the $p$-Laplacian with a variable exponent $p(x,t)$. We prove that if the measurable exponent $p$ is separated from unity and infinity, then the problem has $W$- and $H$-solutions.