Study Of The Blow-up And Asymptotic Behavior Of Solutions To Some Nonlinear Wave Equations

In this thesis,we investigate the properties of the weak solutions to some nonlinear wave equations with damping terms and source terms.The influence of the mechanical behavior between the dissipative term(strong damping or weak damping)and source term(power function source,logarithmic source,variable exponential source)on the blow-up,global existence and asymptotic stability of the solution is discussed.This thesis is divided into five chapters:Chapter 1 is an introduction.First of all,we introduce the background and the current research situation in China and abroad of the problems studied in this thesis.Furthermore,we state the main methods used,main results obtained and innovations in this thesis.Finally,the necessary knowledge used in this thesis is given.In Chapter 2,we investigate the following initial boundary-value problem for wave equation with dissipative term and power function source term(?)where ? is a bounded Lipschitz domain in Rn(n?1)with smooth boundary(?)?,T>0 and u0?H01(?),u1?L2(?),??0,?>-??1,in which ?1 is the first eigenvalue of the operator-? under the homogeneous Dirichlet boundary conditions,and(?)The studies on the blow-up and asymptotic behavior of the solutions to problem(4)were mainly based on the assumption that the initial energy is subcritical and critical,but few results were obtained when the initial energy is supercritical.The main difficulty is that one can not find out the invariant subset similar to Nehari manifold.In this chapter,we construct a new control functional,and give a lower bound estimate of the L2 norm of the solution.Combining with the modified Levine's concave method and energy inequality,the blow-up result of the weak solution in finite time is proved.At the same time,the upper bound of the lifespan is estimated.In addition,the quantitative relation among source term,diffusion term and energy functional is given by constructing a suitable control function,and then the decay estimate of the solution is obtained by using the Komornik inequality.Furthermore,the asymptotic stability of the solution is proved.Finally,we give some numerical simulations to demonstrate our main results.In Chapter 3,the initial-boundary value problem for the following semilinear wave equation with strong damping and logarithmic nonlinear source is considered(?)Here,we always assume that the exponent q satisfies 2<q<+?,if n=1,2;2<q<2*=2n/n-2,if n? 3.Different from the problem(4),logarithmic source is a kind of nonlinear source with special physical background between linear source and power function source.How to analyze its influence on the behavior of the solution is an interesting prob-lem.As is known to all,on the one hand,for the case where the initial energy is supercritical,one can not find out the invariant subset similar to Nehari man-ifold,on the other hand,how to use the logarithmic Sobolev inequality to deter-mine the qualitative relationship between the dissipation term and the source ter-m is a mathematical challenge.We establish the equivalent relationship between the L2 norm of the solution and the energy functional in the sense of correction constants by the qualitative analysis of a new control functional.Furthermore,we prove that the solution blows up in a finite time by using the modified Levine's concave method and some differential inequality techniques.Meanwhile,an upper bound estimate of the lifespan of the weak solution is also obtained.In addition,when q>2n-2/n-2,Sobolev embedding theorem H01(?)?Y2q-2(?)does not hold,and the traditional method of analyzing the lower bound of lifespan of the weak solution fails.In order to overcome these difficulties,we construct a new control functional with a small dissipation term,and then a lower bound estimate of the lifespan of the weak solution is obtained by using energy estimation and differential inequality techniques.In Chapter 4,the initial-boundary value problem for the following quasilinear wave equation with,strong damping and variable exponential source is studied(?)It will be assumed that the exponents m(x),k(x)satisfy the following conditions 2?m-?m(x)?m+<?,1<k-?k(x)?k+<?.When the exponent m(x)is in the interval[m-(1+2n-2m+nm/2n(n-m),nm-/n-m-],Sobolev embedding inequality?u?2m-2?C??u?Lmdoes not hold.So we can not use the method used when m=2 to obtain the lower bound of the lifespan of the weak solution.In order to overcome this difficulty,by means of the interpolation inequality and the energy estimate,we establish an inverse Holder inequality with correction constants for the weak solution to problem(6).Furthermore,we construct a new control functional with a small dissipative term,and then apply the inverse Holder inequality obtained as well as energy estimate to establish a first-order nonlinear differential inequality.Finally,by analyzing the properties of the solution of the differential inequality,a lower bound of the lifespan of the weak solution is obtained.In Chapter 5,we summarize the innovations and main results of this thesis,and state a number of problems to be further researched.