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Abstract
In this paper, we study the global existence of classical solutions for the Cahn-Hilliard equation with terms of lower order and non-constant mobility. Based on the Schauder type estimates, under some assumptions on the mobility and terms of lower order, we establish the global existence of classical solutions. Keywords. Cahn-Hilliard equation, Existence, Uniqueness. AMS Classication: 35K55, 35Q99, 35K25, 82B26
Highlights
We investigate the Cahn-Hilliard equation with terms of lower order
We require some delicate local integral estimates rather than the global energy estimates used in the discussion for the Cahn-Hilliard equation with constant mobility
Since we are concerned with classical solutions, the uniqueness is quite easy by using the standard arguments, and we omit the details
Summary
We investigate the Cahn-Hilliard equation with terms of lower order. on a bounded domain Ω ⊂ R2 with smooth boundary, where k is a positive constant. Only a few papers devoted to the Cahn-Hilliard equation with terms of lower order. It was G.Grun [16] who first studied the equation (1) with degenerate mobility for a special case, namely, A(u) = −u. (3) of the Cahn-Hilliard equation with terms of lower order This means that we should find a new approach to establish the required estimates on u L2(Ω) and ∇u L2(Ω). Our approach is based on uniform Schauder type estimates for local in time solutions To this purpose, we require some delicate local integral estimates rather than the global energy estimates used in the discussion for the Cahn-Hilliard equation with constant mobility.
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