Cauchy Problem Of The Multidimensional Generalized Cubic Double Dispersion Equation

In this paper,we prove that the Cauchy problem of the multidimensional general-ized cubic double dispersion equation vtt-Δv-aΔvtt-bΔ2v-dΔvt=Δf(v), x∈Rn,t>0, (1) v(x,0)=v0(x),vt(x,0)=v1(x), x∈Rn (2) has a unique global solution,where v(x,t)is an unknown function,a,b>0 and d≠0 are constants,△is the n-dimensional Laplace operator and,Δ2 is biharmonic operator, f(s)is a given nonlinear function,v0(x)and v1(x)are given initial value functions.The blow-up of the solution for the Cauchy problem of the multidimensional generalized double dispersion equation is discussed by the concavity method under some conditions.For convenience of discussion,we make scaling transform v(x,t)=u(y,τ) Eq.(1)becomes Without loss of generality,we study the following Cauchy problem utt-Δu-Δutt+Δ2u-αΔut=Δg(u), x∈Rn,t>0, (3) u(x,0)=u0(x),ut(x,0)=u1(x), x∈Rn. (4)The main results are stated as follows:Theorem 1.Suppose that when n=1:2,3,s≥2 and when≥4,s>n/2,u0∈Hs(Rn),u1∈Hs-1(Rn),g∈C[s]+1(R) and g(0)=0,then the Cauchy problem(3),(4) has a unique l0cal solution u∈C([0,T0)；Hs(Rn))∩C1([0,T0)；Hs-1(Rn))∩C2([0:T0)； Hs-2(Rn)),where[0,T0)is a maximal time interval of the existence of the solution Moreover,if then To=∞.In the following,we prove that the extension of the solution(5)for the Cauchy problem(3),(4)transforms the extension condition of the solution(6)below,i.e.,we prove the following theoremTheorem 2.Suppose that when n=1,2,3,s≥2 and when n≥4,s>n/2；u0∈Hs(Rn),u1∈Hs-1(Rn),g∈C[s]+1(R)and g(0)=0,then Cauchy problem(3),(4)has a unique local solution u∈C([0,T0)；Hs(Rn))∩C1([0,T0)；Hs-1(Rn))n∩C2([0,T0)；Hs-2(Rn)), where[0,T0)is a maximal time interval of the existence of the solution.Moreover,if then T0=∞.Lemma 1. Suppose u0∈Hs(Rn),u1∈Hs-1(Rn)(n=1,2,3,s≥2 and n≥4,s≥3/2+n/2),Λ-1u1∈L2(Rn),g∈C[s]+1(R):g(0)=0,and G0(u0)∈L1(Rn).(1)if (?)y∈R,G0(y)≥0,then the solution of Cauchy problem(1.9),(1.10)has the estimation‖Λ-1ut(.,t)‖2+‖u(.,t)‖2+‖ut(.,t)‖2+‖▽u(.,t)‖2≤E(0)e2│α│T,(?)t∈[0,T](T4+n/2,(n=1,2),then the global generalized solutions of the Cauchy problem(3),(4)are the global classical solutions u∈C([0,∞)；Cb4(Rn))∩C1([0,∞)；CB3(Rn))∩C2([0,∞)；CB2(Rn)),n=1,2.Theorem 5.Suppose thatα>0,g(y)∈C(R),u0∈H1(Rn),u1∈L2(Rn),Λ-1u1∈L2(Rn),G0(u)=∫0ug(y)dy,G0(u0)∈L1(Rn)and there exists constantsβ>0,and￡>0,such that 2yg(y)≤2(4β+2+εa)G0(y)+(4β+εα-δ-1α)y2,VΦ∈R, (7) then the generalized or classical solution of the Cauchy problem(3),(4)blows up in finite time if one of the following conditions holds:(1)E(0)<0,(2)E(0)=0,(Λ-1u0,Λ-1u1)+(u0,u1)>0, whereTheorem 6.Suppose thatα>0,g(y)∈C(R),u0∈H1(Rn),u1∈L2(Rn),Λ-1u1∈and there exists constantsβ>0 andε>0,such that 4βε<αand 2yg(y)≤(4β+2+εα)G0(y),Vy∈R, (8) then the generalized or classical solution of the Cauchy problem(3):(4)blows up if one of the following conditions holds: (1)E(0)≤0,...