GLOBAL-IN-TIME SOLUTIONS AND HÖLDER CONTINUITY FOR QUASILINEAR PARABOLIC PDES WITH MIXED BOUNDARY CONDITIONS IN THE BESSEL DUAL SCALE

Hoppe F, Meinlschmidt H, Neitzel I (2024)


Publication Type: Journal article

Publication year: 2024

Journal

Book Volume: 13

Pages Range: 1250-1286

Journal Issue: 5

DOI: 10.3934/eect.2024025

Abstract

We proved the existence and uniqueness of global-in-time solutions in the W−1,p 1,p D-WD 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 Hθ−1,p D = [W−1,p D, Lp ]θ with θ ∈ [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 θ was derived from the maximum degree of Hölder continuity for solutions to an elliptic mixed boundary value problem in Lp.

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APA:

Hoppe, F., Meinlschmidt, H., & Neitzel, I. (2024). GLOBAL-IN-TIME SOLUTIONS AND HÖLDER CONTINUITY FOR QUASILINEAR PARABOLIC PDES WITH MIXED BOUNDARY CONDITIONS IN THE BESSEL DUAL SCALE. Evolution Equations and Control Theory, 13(5), 1250-1286. https://doi.org/10.3934/eect.2024025

MLA:

Hoppe, Fabian, Hannes Meinlschmidt, and Ira Neitzel. "GLOBAL-IN-TIME SOLUTIONS AND HÖLDER CONTINUITY FOR QUASILINEAR PARABOLIC PDES WITH MIXED BOUNDARY CONDITIONS IN THE BESSEL DUAL SCALE." Evolution Equations and Control Theory 13.5 (2024): 1250-1286.

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