OnTwo Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability

摘要

We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term: -△{sup}2u - 1/ε{sup}2 (|u|{sup}2 - l)u = u{sub}t in Ω {is contained in} R{sup}2 or R{sup}3 and -△{sup}2u + (1/ε{sup}2){nabla}·((|{nabla}u|{sup}2 - l){nabla}u) = u{sub}t in Ω {is contained in} R{sup}2 or R{sup}3 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{sub}∞2(0, T, L{sub}2(Ω)) ∩ L{sub}2(0, T,H{sup}2(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 [M.J. Lai, C. Liu and P. Wenston (2004). Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations.