On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability
نویسندگان
چکیده
We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term: −∆2u− 1 22 (|u|2 − 1)u = ut in Ω ⊂ R or R and −∆2u + 1 22 ∇ · ((|∇u|2 − 1)∇u) = ut in Ω ⊂ R or R 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 L∞(0, T, L2(Ω)) ∩ L2(0, T,H(Ω)). 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 in [Lai, Liu and Wenston’03]. AMS 2000 Mathematics Subject Classifications: 35J35, 35K55, 35J60
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