Global <em>W</em><sup>2,<em>p</em></sup> (2<=<em>p</em><infinity) Solutions of GBBM Equations in Arbitrary Dimensions

Global <em>W</em><sup>2,<em>p</em></sup> (2<=<em>p</em><infinity) Solutions of GBBM Equations in Arbitrary Dimensions

Year:    1999

Journal of Partial Differential Equations, Vol. 12 (1999), Iss. 1 : pp. 63–70

Abstract

This paper studies the initial-boundary value problem of GBBM equations  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'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-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; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/1999-JPDE-5525

Journal of Partial Differential Equations, Vol. 12 (1999), Iss. 1 : pp. 63–70

Published online:    1999-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    8

Keywords:    GBBM equation