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Decay of solutions to a water wave model with a nonlocal viscous dispersive term. (English) Zbl 1198.35190

Authors’ abstract: We investigate a water wave model with a nonlocal viscous term
\[ u_t + u_x+ \beta u_{xxx}+ \frac {\sqrt{\nu}} {\sqrt{\pi}} \int_0^t \frac {u_t(s)}{\sqrt{t-s}}\,ds + uu_{x} = \nu u_{xx}. \]
The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74M15 Contact in solid mechanics
76M22 Spectral methods applied to problems in fluid mechanics
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