Research On The Dynamic Behavior Of The Solutions Of Complex Singular Systems

The studies presented in this thesis are from the research projects supported by the National Natural Science Foundation of China "Qualitative study on nonlinear partial differential equations on singular manifolds(11871017)".Real dynamic systems in engineering practice almost always contain a variety of singularity factors.The focus is on two types of singularities that are prevalent in engineering practice,namely,the nonlinear singularities arising from the source term in the system and the geometric singularities contained in the region of the system.Using the potential well theory,this thesis devotes to studying the dependence of the dynamical behaviour of the solution of two singular systems with nonlinear singularity factors or geometric singularity domain factors on initial data,including the initial conditions leading to the global existence and finite time blowup of solutions,and the intrinsic connection between initial data and the dynamical behaviour of solutions,such as global existence,asymptotic,finite time blowup,upper and lower bound estimates of blowup time and blowup rate,etc.Two types of dynamical properties of systems with different non-linear growth properties are considered.Looking at the mathematical models extracted from various engineering problems,it is easy to see that the source terms,as the main nonlinear sources in the models,have a variety of complex mathematical expressions due to their different physical contexts.Based on these,the growth of nonlinear sources can be classified into fast-growing polynomial nonlinear sources and slow-growing logarithmic nonlinear sources:(1)For polynomial-type nonlinear sources,Chapter 2 investigates the global wellposedness of solutions for semilinear hyperbolic equations and parabolic equations with generalized nonlinear sources.The starting point is to find common features between the power-type nonlinear sources and to explore their common role in the two types nonlinear evolution equations,taking as a starting point the ability to encompass a wider and more general form of nonlinearity.To this end,the global well-posedness of the solutions of the two types equations for different initial energy levels is discussed in the same variational framework,and conclusions on the division of initial data for different initial energy levels are given.(2)For logarithmic nonlinear sources,Chapter 3 investigates the global well-posedness of solutions for nonlinear fourth-order dispersive wave equations with logarithmic sources.The complex dispersion-dispersion structures and logarithmic sources for singular system makes the analysis of the blowup more difficult.Therefore,we have improved the potential well theory and obtained the global existence and infinite time blowup of solutions at subcritical and critical initial energy.For blowup at supercritical initial energy,two strategies are adopted to prove the finite-time and infinite-time blowup of the solution at supercritical initial energy by either weakening the dispersion-dispersion structure or enhancing the nonlinearity,respectively.Two types of singular systems with different singularities at domain are considered.The region of the system contains various non-smooth boundaries such as cusps,edges and corners,which usually encounters in engineering practice.Classical mathematica often assumes that the boundary of domain is smooth,which is only simplified.The neglected geometric singularities may cause huge biases in the analysis and calculations for long-time behavior of the system.Typical non-smooth domain contain conical points and wedge-shaped prisms:(3)To address the domain with conical singularities,Chapter 4 studies the global wellposedness of solutions for the nonlinear parabolic equations with singular potential on cone singular manifolds at supercritical initial energy or arbitrary positive initial energy.The singular structure at the boundary of the domain renders classical analytical techniques ineffective and requires to develop theoretical tools and methods based on the singularity structure.Two sufficient conditions leading to finite time blowup of solutions with blowup time at arbitrary positive initial energy are obtained,which reveal the blowup time relying on the initial data,and extend the technical route and ideas for the research on the blowup properties of solutions.(4)To address the domain with edge singularities,Chapter 5 studies the global wellposedness of solutions for the nonlinear pseudo-parabolic equations on wedge manifolds.Compared to conical manifolds,wedge manifolds are less smooth on edges,making it more difficult to develop analysis tools to fit the wedge singular structure.A new inequality combined with the compactness method is used to prove the local existence and uniqueness of the solution,we obtain the global existence,asymptotic behavior and finite time blowup of the solution at subcritical and critical initial energy.At the supercritical initial energy,the finite time blowup of the solution with estimates of upper and lower bound of blowup time are obtained.