Cauchy Problem for a Higher Order Boussinesq Equation

In this thesis, we establish global well-posedness of the following Cauchy problem for a higher-order Boussinesq equation: utt − uxx − uxxtt + _uxxxxtt = (g(u))xx, x 2 R, t > 0 u(x, 0) = '(x), u(x, 0) = (x) where _ is a positive constant, u(x, t) is a real-valued function of two real variables and g(u) is a given function u with g(0) = 0. At the microscopic level the higher order Boussinesq equation was derived in [12] for the longitudinal vibrations of a dense lattice, in which a unit length of the lattice contains a large number of lattice points. Thus, in the above higher order Boussinesq equation which has been written in terms of the scaled variables, u = u(x, t) is the longitudinal strain, t is time and x is a spatial coordinate along the length of the lattice. Our assumptions about the nonlinear term and initial conditions are ' 2 Hs, 2 Hs and g 2 C[s]+1(R) with g(0) = 0. We first establish local existence and uniqueness of the Cauchy problem in the Sobolev space Hs with any s > 1. Under further assumptions on and g, we are able to prove the global existence. Finally, we show continuous dependence of the solution on the initial data