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We study the problem of existence and uniqueness of strong solutions to adegenerate quasilinear parabolic non-Newtonian thin-film equation. Originatingfrom a non-Newtonian Navier--Stokes system the equation is derived bylubrication theory and under the assumption that capillarity is the onlydriving force. The fluid's shear-thinning rheology is described by theso-called Ellis constitutive law. For flow behaviour exponents $\alpha \geq 2$the corresponding initial boundary value problem fits into the abstract settingof [4,Thm. 12.1]. Due to a lack of regularity this is not true for flowbehaviour exponents $\alpha \in (1,2)$. For this reason we prove an existencetheorem for abstract quasilinear parabolic evolution problems with H\"oldercontinuous dependence. This result provides existence of strong solutions tothe non-Newtonian thin-film problem in the setting of fractional Sobolev spacesand (little) H\"older spaces. Uniqueness of strong solutions is derived byenergy methods and by using the particular structure of the equation.

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation