The Cauchy Problem Of A System For A Class Of The Nonlinear Wave Equations With Several Variables

This paper consists of three chapters. The first chapter is the introduction. The origin of the model equation of the study in this paper, some known results and the notations are given.In the second chapter,we study the following Cauchy problem for a system of the n-dimensional nonlinear wave equations. utt(x,t)-σΔu(x,t)-Δutt(x,t)=Δf(v(x,t)), x∈Rn, t>0, (1) vtt(x,t)-Δvtt(x,t)=Δg(v(x,t)), x∈Rn, t>0, (2) u(x,0)=u0(x), ut(x,0)=u1(x), x∈Rn, (3) v(x,0)=v0(x), vt(x,0)=v1(x), x∈Rn, (4)where u(x,t) and v(x,t) are the unknown functions;Δdenotes the n-dimensional Laplace opertor andσ>0 is a constant; f(y) and g(y) are given nonlinear functions; u0(x), u1(x), v0(x), and v1(x) are given initial value functions defined on Rn. For this purpose, the problem (1)-(4) can be written the following vector form: Wtt-ΔWtt=ΔF(u,v), x∈Rn, t>0, (5) W(x,0)=W0(x), Wt(x,0)=W1(x), x∈Rn, (6) whereThen, using the contraction mapping principle,we prove the Cauchy problem (5), (6) has a unique global generalized solution in C2([0,∞); Hs(Rn)×Hs(Rn)), and a unique global classical solution in C2([0;∞); Hs(Rn)×Hs(Rn)). The main results are as follows:Theorem 1 Suppose that s>n/2, W0,W1∈Hs(Rn)×Hs(Rn),f∈C[s]+1(R),f(0)= 0,g∈C[s]+1(R),g(0)=0. Then the Cauchy problem (5), (6) admits a unique local generalized solution W∈C2([0,T0), Hs×Hs), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Now, we prove that the extension condition of the solution (7) for the Cauchy problem (5), (6) transforms the extension condition of the solution (8) below, i.e, we prove the following theorem.Theorem 2 Suppose that s>n/2, W0,W1∈Hs×Hs,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0. The the Cauchy problem (5), (6) admits a unique local generalized solution W∈C2([0,T0),Hs×HS), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Theorem 3 Suppose that s≥3/2+n/2, W0, W1∈Hs×Hs,Λ-1W1∈L2×L2,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0,z(v)=∫0vg(y)dy≥0. If there existγsatisfies such that where A, B>0 are constants, then the Cauchy problem (5) and (6) admits a unique global generalized solution W∈C2([0,∞), Hs×Hs). Theorem 4 Suppose that W0, W1∈Hs×Hs, v1,Λ-1v1∈L2, g∈C(R), z(v0)∈L1,f(v)=v, and whereα>0 is a constant. Then the generalized solution or the classical solution W(x, t) of the Cauchy problem (5), (6) blows up in finite time if one of the following conditions holds(1) E(0)<0,(2) E(0)=0,<Λ-1v1,Λ-1v0>+>0,In Chapter 3, we prove that the Cauchy problem (5), (6) has a unique global gen-eralized solution in C3([0,∞); (Wm,p∩L2)×(Wm,p∩L∞∩L2)) when m≥0 and a unique global classical solution in C3([0,∞); (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)) when m>2+n/p. The main results are as follows:Theorem 5 Suppose that W0, W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+1(R)(m≥0), andf(0)=0,g(0)=0. Then the Cauchy problem (5) and (6) admits a unique local generalized solution W∈C2([0,T0);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))(m≥0), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Theorem 6 Suppose that(1)WO, W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+1(Rn)(m≥0). (2)f, g∈Cm+1(R),f(0)=0,g(0)=0, and z(v)>0, ifγsatisfies such that where A,B>0 are constants, the Cauchy problem (5), (6) admits a unique global generalized solution W∈C3([0,∞); (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)).Lemma 1 Suppose that the conditions of Theorem 6 hold and f, g∈Ck+m+1(R), where k≥0 is arbitrary integer. Then the generalized solution W(x,t) belongs to Ck+3+l([0,T]; (Wm-l,p∩L∞∩L2)×(Wm-l,p∩L∞∩L2)), (VT>0),0≤l≤m.Theorem 7 Suppose that the conditions of Lemma 1 hold and k=0, l=0, m> 2+n/p, then the Cauchy problem (5), (6) has a unique global classical solution W(x, t)∈C3([0,T]; (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)), i.e., W(x,t)∈C3([0,T]; (CB2(Rn)∩L∞∩L2)×(CB2(Rn)∩L∞∩L2)), where CB2(Rn) consists of all those functions in C2(Rn) that are bounded in Rn.