Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane

Jerry L. Bona; S. M. Sun; Bing-Yu Zhang

Displaying similar documents to “Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane”

Comparison of KP and BBM-KP models.

Daspan, Gideon P., Tom, Michael M. (2007)

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Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order

Guo Wang Chen, Shu Bin Wang (1995)

Commentationes Mathematicae Universitatis Carolinae

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The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation $u_{t t} - α u_{x x} - β u_{x x t t} = ϕ {(u_{x})}_{x}$ are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods $u_{t t} - [a_{0} + n a_{1} {(u_{x})}^{n - 1}] u_{x x} - a_{2} u_{x x t t} = 0 .$

On KP-II type equations on cylinders

Axel Grünrock, Mahendra Panthee, Jorge Drumond Silva (2009)

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Tomoeda, Kyoko (2011)

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Unique local existence of solution in low regularity space of the Cauchy problem for the mKdV equation with periodic boundary condition

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On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock (2010)

Open Mathematics

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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces ${\hat{H}}_{s}^{r} (ℝ)$ defined by the norm ${∥v_{0}∥}_{{\hat{H}}_{s}^{r} (ℝ)} : = {∥{〈ξ〉}^{s} \hat{v_{0}}∥}_{L_{ξ}^{r^{'}}}, 〈ξ〉 = {(1 + ξ^{2})}^{\frac{1}{2}}, \frac{1}{r} + \frac{1}{r^{'}} = 1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac{2 j - 1}{2 r^{'}}$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $\frac{2 j - 1}{2 r^{'}}$ . The results for r =...