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In the present article we first study the existence of the stationary solution to an initial boundary value problem for the Mullins equation of fourth order, which was proposed by Mullins [“Two-dimensional motion of idealized grain boundaries,” J. Appl. Phys. 27, 900 (1956)] to describe the groove development, due to the surface diffusion, at the grain boundaries of a heated polycrystal. Then employing an energy method we prove that this stationary solution is asymptotically stable in a suitable norm as time goes to infinity.

Asymptotic stability of the stationary solution to an initial boundary value problem for the Mullins equation of fourth order