Research On The Well-posedness Of The Wave Equation Solution With Time-dependent Damping Terms

This paper uses the theory of potential well to study the dynamic behavior of wave equations with the coefficients of weak damping terms about time and nonlinear wave equations with scale-invariant damping and mass,mainly focusing on the dependence of the initial value on the solution.According to the relationship between the depth of potential well and the initial energy,the well-posedness of solutions are studied.The second chapter studies the initial-boundary value problem of the wave equation with the coefficients of weak damping term about time.First,the Galerkin method is used to obtain the existence and uniqueness of the solution for corresponding linear problem,and then the local solution of the problem is obtained by the contraction mapping principle.Then using the contradiction method,the stable or unstable sets under three different energy levels are obtained,and by combining the local solution theorem and the boundedness of the solution,the global existence theorem of the solution under the low-initial-energy condition is obtained.And the finite time blow up of the solution is obtained by the concave function method.Finally,based on the concave function method and the control of the indexes and coefficients of the time-related terms,the finite time blow up of the solution at the supercritical energy level is derived.The third chapter studies the initial-boundary value problem of wave equations with scale-invariant damping and mass.After defining the weak solution,the existence and uniqueness of the local solution is first given in this chapter,and then the properties of several types of functionals are used to obtain the depth of potential well.Through the control of scale-invariant weak damping and quality-related terms,the existence and nonexistence of global solution at low and critical initial energy levels as well as the finite time blow up of the solution at supercritical energy level are obtained.