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Here, $\\Omega \\subset \\mathbb{R}^N$, $N \\geq 2$, is a bounded domain, $T>0$, \\[ \\mathcal{F}(z,\\nabla u)=a(z)|\\nabla u|^{p(z)-2} + b(z) |\\nabla u |^{q(z)-2} \\] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. 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