# Moscow Mathematical Journal

Volume 16, Issue 1, January–March 2016  pp. 179–200.

A Uniform Coerciveness Result for Biharmonic Operator and its Application to a Parabolic Equation

Authors:  Kazushi Yoshitomi
Author institution: Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minamiohsawa 1-1, Hachioji, Tokyo 192-0397, Japan

Summary:

We establish an L2 a priori estimate for solutions to the problem: ∆2u = f in Ω, ∂u/∂n = 0 on ∂Ω, −∂/∂n(∆u) + βαu = 0 on ∂Ω, where n is the outward unit normal vector to ∂Ω, α is a positive function on ∂Ω and β is a nonnegative parameter. Our estimate is stable under the singular limit β → ∞ and cannot be absorbed into the results of S. Agmon, A. Douglis and L. Nirenberg. We apply the estimate to the analysis of the large-time limit of a solution to the equation (∂/∂t+∆2)u(x,t) = f(x,t) in an asymptotically cylindrical domain D, where we impose a boundary condition similar to that above and the coefficient of u in the boundary condition is supposed to tend to +∞ as t → ∞.

2010 Math. Subj. Class. 35J35, 35J40, 35K35.

Keywords: Biharmonic operator, singular perturbation, parabolic equation, stabilization.