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This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth: \[ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z) |\nabla u|^{q(z)-2} \nabla u \right) = F(z,u) \quad \text{in $Q_T=\Omega \times (0,T)$} \] where $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain with the boundary $\partial\Omega\in C^2$, $z=(x,t)\in Q_T$, $a:\bar Q_T \mapsto \mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term has the form \[ F(z,v)=f_0(z)+b(z)|v|^{\sigma(z)-2}v. \] The variable exponents $p$, $q$, $\sigma$ are given functions defined on $\bar{Q}_T$, $p$, $q$ are Lipschitz-continuous and \[ \dfrac{2N}{N+2}0. \] |
