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Zbl 1145.35031
Escher, Joachim; Yin, Zhaoyang
Initial boundary value problems of the Camassa-Holm equation. (English)
[J] Commun. Partial Differ. Equations 33, No. 3, 377-395 (2008). ISSN 0360-5302; ISSN 1532-4133/e

Summary: We study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa-Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa-Holm equation on a compact interval possesses no nontrivial global classical solutions.

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35G25 Initial value problems for nonlinear higher-order PDE
35L05 Wave equation
35Q35 Other equations arising in fluid mechanics

Keywords: blow-up and global existence; global weak solutions; local well-posedness; the Camassa-Holm equation; one space dimension; conservation of symmetry

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Zentralblatt MATH Berlin [Germany]