On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability

Abstract

We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term:

and

with an initial value and a Dirichlet boundary conditions. We show the existence and uniqueness of the weak solutions of these two equations. For any t ∈ [0, + ∞ ), we prove that both solutions are in . We also discuss the asymptotic behavior of the solutions as time goes to infinity. This work lays the ground for our numerical simulations for the above systems [M.J. Lai, C. Liu and P. Wenston (2004). Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations. Applicable Analysis, 83, 563–577].

Keywords:

• Nonlinear biharmonic evolution equations
• Ginzburg–Landau type equations
• Liquid crystals
• Existence and uniqueness
• Asymptotic behavior

Acknowledgment

Ming-Jun Lai was supported by the National Science Foundation under Grant #DMS-9870178.

Notes

^‡E-mail: [email protected]

^†E-mail: [email protected]

^†E-mail: [email protected]

Additional information

Notes on contributors

Paul Wenston ‡

‡E-mail: [email protected] †E-mail: [email protected]