Theoretical Research On Nonlinear Dynamic Systems

The research topics of the present thesis are from the research projects sup-ported by the National Natural Science of China,which are"Several classes ofnonlinear physical mathematics model equations and parabolic equations"(No.10271034) and"Dynamics of high order evolution equations and Schr(o|¨)dingerequations"(No.10871055).This work makes qualitative studies on severalnonlinear dynamical systems by using potential wells method.The invariantsets and vacuum isolations of solutions to the relative mathematical physicsproblems are obtained.And the well-posedness and finite time blows up of so-lutions to the systems are studied.Especially the sharp conditions(thresholdresults) for global existence are given.For some special problems,the long timebehaviors are concluded.First,the thesis studies the initial boundary value problem for a classof nonlinear wave equations with several nonlinear terms,which is a generalform of Klein-Gordon equation from quantum mechanics.The nonlinearityunder consideration reflects the affects of several nonlinear sources to the well-posedness of the system,without any restriction on the number of the sourceterms.By constructing the variational problem this work derives the sufficientand necessary conditions for global existence of the problem and discusses thecritical cases only for global existence.This work considers the several sourceterms with different signs further.The difficulties of energy estimates for non-positive energy are overcame by constructing the potential well theory.Thethesis clarifies some properties of some relative functionals of potential wells,and gives the relations between these functionals and a special ball.Also thedepth function of potential wells is described for this case of nonlinearities withdifferent signs.The sufficient and necessary conditions for global existence and non-existence are shown,also are the vacuum isolation and the global case forcritical condition.For the fact that the nonlinear reaction-diffusion equationpossesses a similar potential energy expression to that of wave equation,thiswork obtains the parallel results to those of wave equations.Then,this thesis considers the system equation with complex structure.First this thesis focuses on the initial boundary value problem for a class offourth-order wave equations with dissipative and nonlinear strain terms,whicharose from weakly nonlinear analysis of elasto-plastic-microstructure models.For this problem,the collapse for positive energy case and the well-posednessfor critical condition are both primarily concerned problems.Based on the fun-damental theory of variational problem,in aid of the techniques like integrationestimations,equivalent norms,integration substitution and so on,this thesisintroduces the new potential wells for this class of high order complex nonlin-ear models and gets the properties of the potential wells.And the relationsbetween these functionals and a ball in H₀²(Ω) are pointed out.In this thesis,it is the first time to set the variational problem for the positive derivativesspace and study the relations with the original variational problem.By divid-ing the space according to positive and negative derivatives,it is pointed outthat the minimizer belongs to Nehari manifold of positive derivatives space.Then this work estimates the depth of a single potential well and obtains theinvariant sets of the problem and vacuum isolation.Then the thesis gives thesharp conditions(threshold results) of global existence and finite time blow upof solutions for positive energy.These results parallel the outside and inside ofthe ball in H₀²(Ω) space.For the cases that initial energy is greater than zero,equal to zero or less than zero,the system blow up is proved under variousassumptions.For the critical energy,a comprehensive consideration is taken for the global existence and non-existence of solutions,and the sharp conditionis obtained.Then some assumptions are simplified and made more concrete inorder to be applied in engineering practices.Also some special valid functionsin engineering problems are given as examples.For the initial boundary valueproblems of a class of fourth order wave equations with dispersive term anddissipative term and a class of strongly damped nonlinear wave equations re-spectively,by using multiplier method the corresponding asymptotic behaviorsof the global solutions are obtained.Further,the thesis studies a class of generalized Boussinesq equations.Boussinesq-type equations were introduced to describe the motion of waterwave with small-amplitude long waves,which are frequently used in computermodels for the simulation of water waves in shallow seas and harbors.For theCauchy problem of a class of generalized Boussinesq-type equations,the presentthesis discusses the global existence and non-existence of the open problemswith f(u) =±|u|^p and f(u) =-|u|^p-1(p＞1).First,by applying Fouriertransformation the energy conservation is obtained.Then for both positiveenergy and non-positive energy the thesis gives some properties for potentialwells and derives the vacuum isolation of solutions.Based on above derivedproperties,this thesis proves the sharp condition of global-in-time existenceand blow up of solutions to above problems.At the critical energy level,thesimilar sharp condition of global well-posedness of above problems is also ob-tained.Differing from the dissipative system,this sharp condition relies on therequirement of positive inner product of initial data.Finally the present thesis simulates and analyzes the characters of the po-tential energy functionals,the initial data and the depth of potential wells.Alsothe simulations show how the complex source terms affect the above problems.