# 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 *L*^{2} a priori estimate for solutions to the problem: ∆^{2}*u* = *f* in Ω, ∂*u*/∂*n* = 0 on ∂Ω, −∂/*∂n*(∆u) + βα*u* = 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.

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