(untitled) (Journal of Partial Differential Equations, 2020)
In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_∂Ω = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)|f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ 2n-2 if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^2,p(Ω) ∩ W^2,2(Ω) ∩ W^1,p_0(Ω)(W^2,p(R^n) ∩ W^2,2(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^1,∞(0, T; W^2,p(Ω) ∩ W^2,2(Ω) ∩ W^1,p_0(Ω))(W^1,∞(0, T; W^2,p(R^n) ∩ W^2,2 (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.