In this paper, we study the weak solutions of a sixth-order phase-field equation with degenerate phase-dependent diffusion mobility in 2D case. The main features and difficulties of this equation are given by a highly nonlinear sixth-order elliptic term, a strong constraint imposed by the presence of the nonlinear principal part and the lack of maximum principle. Based on the Schauder-type estimates and entropy estimates, we are able to prove the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions by using approximation and compactness tools. In the end, we study the nonnegativity of solutions for the sixth-order degenerate phase-field equation.
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The authors are indebted to the referees for careful reading of the paper and helpful suggestions. This work is partially supported by “Fundamental Research Funds for the Central Universities (Grant No. JUSRP116030),” “Natural Science Foundation of China for Young Scholar (Grant No. 11401258)” and “Natural Science Foundation of Jiangsu Province for Young Scholar (Grant No. BK20140130).”
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