On The Long Time Well-posedness Of Classical Solutions To Nonlinear Wave Equations

In this thesis,we are concerned with the long time well-posedness of smooth small data solutions to the Cauchy problem for n dimensional quasi-linear wave equations(?) and for the following initial-boundary value problem(?) where u=2,3,(?)stand for the time-space derivatives and the d'Alembertian operator,respectively,O=Rn\K and K is a compact convex obstacle with smooth boundary,v is the unit outer normal vector of(?)and ?>0 is a small parameter.The Einstein summation convention is employed throughout the thesis,that is,the repeated Greek index ?,?,?,...mean the summation from 0 to n.Since we consider the smooth solutions to the problems(0.0.7)and(0.0.8),Q??((?)u),S((?)u)and the initial data(f,g)should be assumed to be smooth in their arguments re-spectively.In addition,Q??((?)u)satisfies the symmetry condition Q??((?)u)=Q??((?)u)for any ?,?=0,1,···,n.Furthermore,we suppose that Q??(0)=S(0)=S'(0)=0,for any ?,?=0,1,…,n and they respectively have the Taylor expansion near(?)u=0,(?) and(?) where N0??=N0??,N1???=N1???,N2???? and N3???=N3??? are constants.We introduce the following notations N0(?):=N0??????.N1(?):=N1?????????,N2(?):=N2????????????,N3(?):=N3?????????,where ?=(?0,…,?n)with ?0=-1,?i=xi/|x|,i=1,2,…,n.We say that the null conditions are satisfied by the quadratic nonlinearity of(0.0.7)or(0.0.8),if the coefficients in(0.0.9)and(0.0.10)fulfill Ni(?)=0,i=0,1.(0.0.11)We say that the null conditions are satisfied by the cubic nonlinearity of(0.0.7)or(0.0.8),if the coefficients in(0.0.9)and(0.0.10)fulfill Ni(?)?0,i=2,3.(0.0.12)The main results of this thesis can be stated briefly as follows:In Chapter 2,we are devoted to exploring another method of[3],in which there is not the semilinear term S((?)u)in(0.0.7)(see[14]for the case with the term S((?)u)).In[3]and other related two dimensional work,the "ghost weight" technique is a key ingredient.However,in this chapter,by establishing the estimates of flux through the outgoing cone instead of the "ghost weight" technique,we also prove the global existence or almost global existence of smooth small data solutions for the Cauchy problem(0.0.7)with n=2 when both(0.0.11)and(0.0.12)hold,or when only(0.0.11)is valid,respectively.The proofs of both[3]and Chapter 2 are dependent on the compactness of the support of initial data.When the property of the compactness is removed,in[17],the authors considered the Cauchy problem for two dimensional nonlinear wave equations with nonlinearities depending only on the derivatives of the unknown function and proved that the smooth small data solutions to this kind of problems exist globally,provided that both the quadratic and cubic nonlinearities fulfill the null conditions.If the null conditions hold only for the quadratic nonlinearities in this situation,we give out the proof of the almost global existence of small data solutions to the problems in Chapter 3.To achieve the aim,we will build up the decay estimates for L?-L?norm of smooth solutions to nonlinear wave equations rather than the usual L?-L2 decay estimates.Besides,a short proof of the global existence of smooth small data solutions to the problems in three dimensional case is also given when only the quadratic nonlinearities satisfy the null conditions.In Chapter 4,we concentrate on considering the three dimensional mixed problem(0.0.8)in exterior domains with Neumann boundary conditions.We prove that the classical solutions to the mixed problem(0.0.8)exist almost globally when(0.0.11)fails.Compared with Dirichlet problem[22]and[35]and Neumann problem[29],there are some essential difficulties to overcome.In[22]and[35],the boundary terms arising from the L2 energy estimates provide a favorable sign and it is beneficial for closing the energy estimates.However,it is sure that this kind of the favorable sign can not be found in the Neumann problem(0.0.8).In[29],the nonlinearity usually has a better decay since(0.0.11)holds.As the acceptable price,it is necessary to estimate the L2 norm of(?)with m>2.In the current chapter,we have to use the trace theorem to treat the boundary terms and the sharp Huygens principle to close the energy estimates.At last,we find that this idea can be realized using at most one scaling operator L.