Persistence Properties And Unique Continuation Of Strong Solutions Of Shallow Water Equations

We consider persistence properties and unique continuation of strong solu-tions of the generalized Camassa-Holm equations and a two-component Camassa-Holm system. A solution is persistent if exponential decay of its initial value to-gether with its spacial derivative implies the same at any latter time. The uniquecontinuation property holds if exponential decay of its initial value together withits spacial derivative and exponential decay at any later time of a solution im-plies that it is trivial. In particular, a strong solution of the Cauchy problem withcompact initial profile cannot be compactly supported at any later time unless itis the zero solution.In this paper, it is shown that the persistence properties of the strong solu-tions of the generalized Camassa-Holm equation and a two-component Camassa-Holm system hold. Furthermore for a subclass of the generalized Camassa-Holmequation considered here, unique continuation of strong solutions hold.This article is divided into 6 chapters. Chapter 1 is introduction. Prelimi-naries needed in the proofs of the conclusions are given in Chapter 2. In Chapter3, we prove the persistence properties of strong solutions of a two-componentCamassa-Holm system. In Chapter 4, we prove the local well-posedness of strongsolutions of the generalized Camassa-Holm equation. In Chapter 5, we prove thepersistence property and unique continuation of strong solution of the general-ized Camassa-Holm equation. Finally, we conclude in Chapter 6 and give somecomments on possible future work.