| In this thesis,we axe concerned with the blowup criteria and the existence and uniqueness of global weak solutions for the dissipative Dullin-Gottwald-Holm equation:Eq.(0.1)describes the unidirectional propagation of surface waves in shallow water regime which was derived by the method of asymptotic analysis and a near-identity normal form transformation from water wave theory,combining the linear dispersion of the KdV equation with the nonlinear dispersion of the Camassa-Holm equation.In the first part of the thesis,we consider the 1 Local-in-space’ blowup criteria for Eq.(0.1).tLocal-in-space,blowup criteria is that uO is local at the point x0 where the negative infimum is achieved.This means that it is impossible to prevent the blowup without modifying uO around the point xO:perturbing u0 only outside a neighborhood of x0 may just help in delay the formation of singularities.Two’local-in-space’ blowup criteria are obtained by delicate analysis on the evolution of some linear combinations of the solution u and its derivative ux.Theorem 1.0.2.Let λ=0,c0+γ/α1=0 and u0∈Hs(R),s>3/2.Suppose that there exists a point X1 ∈R such that u0,x(x1)<-1/α|u0(x1)|.Then the corresponding solution u(t,x)to Eq.(0.1)blows up in finite time with an estimate of the blowup time T*as 0<T*≤(?)Theorem 1.0.3.Let λ>0 and u0 ∈Hs(R),s>3/2.Suppose that there exists a point x2 ∈R such that u0,x(x2)<-λ-1 1/α|u0(x)± λα-(?)C for α≤1 and u0,x(x2)<-λ-1/α|u0(x2)±λα|-(?)C for α>1,whereC=(?)Then the corresponding solution u(t,x)to Eq.(0.1)blows up in finite time with an estimate of the blowup time T*as 0<T*≤(?)for α≤1 and 0<T*≤(?)for α>1Remark:Theorem 1.0.2 generalizes Corollary 2.3 obtained by L.Brandolese in[2](see P3,Remark 1.0.1).Theorem 1.0.3 improves the main result-Theorem 1 in[27](see P4,Remark 1.0.2).In the second part of the thesis,we consider the existence and uniqueness of global weak solutions to Eq.(0.1).The definition of weak solution is given in P6,Definition 1.0.1.Our method relies on the approximation of the initial data u0 ∈ H1(M)by smooth functions,which produces a sequence of global solutions to Eq.(0.1)in H3(R).Suitable priori estimates enable us to extract a subsequence of these solutions that converges weakly in H1(R).The global weak solution result is as follows.Theorem 1.0.4.Let γ=-c0α2 and λ≥ 0.Assume that u0 ∈H1(R),and y0:=(u0-u0,xx)∈M(R).Assume further that there exists x0∈R such that suppy0-(?)(-∞,x0)and suppy0+(?)(x0,∞).Then Eq.(0.1)has a unique weak solution u∈Wloc 1,∞(R+× R)∩Lloc∞(R+;H1(R))with initial data u(0)=u0,moreover the unique weak solution u ∈ C(R+;H1(R))∩ C1(R+;L2(R))and y(t,·):=(u(t,·)-α2uxx(t,·))∈Lloc∞(R+;M(R)).Remark:Theorem 1.0.4 generalizes the main result-Theorem 3.1 obtained in[31](see P6,Remark 1.0.4).The thesis consists of four parts.An introduction to some physical background and main results about the dissipative Dullin-Gottwald-Holm equation is presented in the first chapter.In Chapter 2,we will give some preliminaries and lemmas for the proof of the main results.The blowup results are proved in Chapter 3 and the proof of global weak solutions is given in the last chapter. |
