(untitled) (Journal of Partial Differential Equations, 1999)
This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) (a) u(x, 0) = u_0(x)(b) u |∂Ω = 0 (c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ 2n-2 if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^2⋅p(Ω) ∩ W^1⋅p_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^2⋅p solution u ∈ W^1,∞(0, T; W2⋅p(Ω)∩ W^1⋅p_0(Ω)), so the known results are generalized and improved essentially.