Qualitative Study Of A Class Of Non-local Dispersive Wave Equations

In this dissertation,we study the one-dimensional modified Euler-Poisson system and the two-component Fornberg-Whitham system qualitatively.In Chapter 1,we give the research background,research status and preliminary knowledge of the one-dimensional modified Euler-Poisson system and the two-component Fornberg-Whitham system.In Chapter 2,we investigate the existence of a weak solution to a modified 1-D Euler-Poisson system in the lower order Sobolev space H^s（R）×H^s-1（R）(s∈（1,3/2]）.As the absence of some useful conservation laws,First,we use the improved pseudo-parabolic regularization method to establish the local well-posedness of the solutions of the approximate equations.Then the existence of a weak solution is established by the Aubin-Lions lemma and the weak convergence property.In Chapter 3,we show the existence of a weak solution and wave-breaking for a strong solution to a two-component Fornberg-Whitham system.we first establish the existence of a weak solution to the system in the lower order Sobolev space H^s（R）×H^s-1（R）(s∈（1,3/2]）via a similar method applied in Chapter 2.And then,a blow-up scenario for a strong solution to this system is shown.Finally,we present some sufficient conditions on the initial data that lead to the blow-up for the corresponding strong solution to the system,and suggest that the blow-up for the system may occur even with small slope of the initial value.