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On the uniqueness and large time behavior of the weak solutions to a shallow water equation. (English) Zbl 1034.35115
The authors deal with the uniqueness and larger time behaviour of weak solutions to the Cauchy problems for the following one-dimensional shallow water equation $$\cases \partial_t u+u\partial_x u+\partial_xP=0,\ t>0,\ x\in\bbfR^1\\ P(t,x)=\tfrac 12 \int^\infty_{-\infty} e^{-\vert x-y\vert} \Bigl(u^2+\tfrac 12 (\partial_x u)^2\Bigr) (t,y)dy\\ u(0,x)=u_0(x)\in H^1(\bbfR^1),\endcases$$ which is formally equivalent to the Camass-Holm equation $$\partial_tu-\partial_x^2 \partial_tu+ 3u\partial_x u=2\partial_x u\partial_x^2u+u \partial_x^3u.$$ Moreover, the authors show that the admissible weak solutions (under some additional condition on the solutions) tend to 0 pointwisely as $t\to\infty$.

35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
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