Abstract
We consider evolutionary problems associated with a convex integrand \({f : \Omega_T \times {\mathbb R}^{Nn} \to [0, \infty)}\), which is \({\alpha}\)-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions \({u : \Omega_T \to {\mathbb R}^N}\), which solve
weakly in \({\Omega_{T}}\). Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely
For
we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.
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Singer, T. Existence of weak solutions of parabolic systems with p, q-growth. manuscripta math. 151, 87–112 (2016). https://doi.org/10.1007/s00229-016-0827-1
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DOI: https://doi.org/10.1007/s00229-016-0827-1