TY - JOUR
T1 - Global <em>W</em><sup>2,<em>p</em></sup> (2<=<em>p</em><infinity) Solutions of GBBM Equations in Arbitrary Dimensions
JO - Journal of Partial Differential Equations
VL - 1
SP - 63
EP - 70
PY - 1999
DA - 1999/12
SN - 12
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5525.html
KW - GBBM equation
KW - initial-boundary value
KW - global W^{2⋅p} solution
AB -  This paper studies the initial-boundary value problem of GBBM equations &nbsp;u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f&#39;(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ &lt; ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p &lt; ∞), then ∀T &gt; 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.