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We investigate existence and regularity of weak solutions of a 1-dimensionalparabolic differential equation with a non-constant H\"older diffusioncoefficient and a rough forcing term. Such an equation appears in studying the1-dimensional Ericksen-Leslie model for nematic liquid crystals where ourresult applies. The result presented here uses the H\"older continuity of thediffusion coefficient which comes from the physical background and the analysisof the Ericksen-Leslie model. Moreover, the dependence of the H\"older exponentof the solution is explicit on the H\"older exponent of the diffusioncoefficient.

On parabolic partial differential equations with H\"older continuous diffusion coefficients