| The shallow-water wave equation has been a hot research topic in Partial Differential Equations It is ubiquitous in our lives and is used to model fluids in the oceans and atmosphere,which has led scientists to further their study of the shallow water wave equation.we will consider the blow-up,the well-posedness of the solutionand the infinite propagation speed of three nonlinear shallow water wave systems.Firstly,we prove the weak well-posedness of two-component b family systems(b2)when the initial value(u0,ρ0)∈H1(R)∩W1,∞(R)×L2(R)∩L∞(R)by using the characteristic line method.Based on the relationship between the two-component b family systems and corresponding ODE system,we obtain the existence and uniqueness of the local weak solution of two-component b family system Moreover the stability of the solutions with respect to the initial value is given.Then,we consider the infinite propagation speed and blow-up of the general two-component Dullin-Gottwald-Holm system(GDGH2).More precisely,a new sufficient condition which ensure strong solutions blow-up is given,and it is shown that for nontrivial classical solutions of GDGH2,the initial values have compact supports cannot guarantee that the corresponding solutions have compact supports.Finally,by means of the transport theory and the Littlewood-Paley theory,we establish the local well-posedness of two-component b family system(Rb2)under the action of Coriolis forces which is caused by the rotation of the earth in Besov space Bp,rs × Bp,r,-1,s>max{1+1/p,3/2,2-1/p},p,r ∈[1,∞).It is helpful to study the well-posedness,wave-breaking and other related properties of the shallow water wave equations for a deep understanding of shallow water waves. |
