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Persistence properties and unique continuation of solutions of the Camassa-Holm equation. (English) Zbl 1142.35078
The authors study the Camassa-Holm equation
\[ u_t - u_{txx} + 3uu_x -2u_x u_{xx} - u u_{xxx} = 0, \]
which physically was derived as a shallow water equation admitting peaked solitons. They prove that, given a strong solution to the Cauchy problem for this equation such that the initial data \(u(x,0)\) decays exponentially together with the spatial derivative, if at some time \(t_1>0\) the solution \(u(x,t_1)\) decays exponentially in \(x\) then \(u\) is identically zero: \(u(x,t) \equiv 0\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35B60 Continuation and prolongation of solutions to PDEs
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