Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent

The paper is concerned with the solvability of the initial-boundary value problem for second-order parabolic equations with variable nonlinearity exponents. In the model case, this equation contains the $p$-Laplacian with a variable exponent $p(x,t)$. The problem is shown to be uniquely solvable, provided the exponent $p$ is bounded away from both $1$ and $\infty$ and is log-Hölder continuous, and its solution satisfies the energy equality.
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