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Global-in-time solutions and Hölder continuity for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale

  • *Corresponding author: Hannes Meinlschmidt

    *Corresponding author: Hannes Meinlschmidt 

This research was carried out while F.H. was affiliated with University of Bonn and partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Projektnummer 211504053–SFB 1060. I.N. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Projektnummer 211504053–SFB 1060

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  • We proved the existence and uniqueness of global-in-time solutions in the $ W^{-1,p}_D $-$ W^{1,p}_D $ setting for abstract quasilinear parabolic PDEs with nonsmooth data and mixed boundary conditions, including a nonlinear source term with at most linear growth. Subsequently, we used a bootstrapping argument to achieve improved regularity, in particular Hölder continuity, of these global-in-time solutions within the functional-analytic setting of the interpolation scale of Bessel-potential dual spaces $ H^{\theta-1,p}_D = [W^{-1,p}_D,L^p]_\theta $ with $ \theta \in [0,1] $ for the abstract equation under suitable additional assumptions. This was done by means of new nonautonomous maximal parabolic regularity results for nonautonomous differential operators with Hölder-continuous coefficients on Bessel-potential spaces. The upper limit for $ \theta $ was derived from the maximum degree of Hölder continuity for solutions to an elliptic mixed boundary value problem in $ L^p $.

    Mathematics Subject Classification: Primary: 35A01, 35K59; Secondary: 35R05, 35B65.


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