Nash-Moser techniques for nonlinear boundary-value problems.

Poppenberg, Markus

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Global existence of solutions of parabolic problems with nonlinear boundary conditions

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Global solutions via partial information and the Cahn-Hilliard equation

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Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.

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Existence of weak solutions and an $L_{\infty}$ -estimate are shown for nonlinear nondegenerate parabolic systems with linear growth conditions with respect to the gradient. The $L_{\infty}$ -estimate is proved for equations with coefficients continuous with respect to x and t in the general main part, and for diagonal systems with coefficients satisfying the Carathéodory condition.