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Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions

Sahbi Boussandel — 2018

Applications of Mathematics

The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p -Laplace operator.

Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Sahbi BoussandelRalph ChillEva Fašangová — 2012

Czechoslovak Mathematical Journal

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and L 2 -maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...

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