Global Well-posedness For Some Wave Equations Of Kirchhoff Type

This work carries out qualitative studies on hyperbolic equations of Kirchhoff type withdamping and nonlinear source terms in a bounded domain. We study the initial boundary valueproblem concerning this system. Characterizations with respect to qualitative properties of thesolution in a bounded domain under three energy levels of E(0): global existence, blow-up, asymptotic behavior. In particular, in finite time blow up result with arbitrary high initialenergy E(0)>0is given in this article. This work is based on potential wells theory, concavitymethod and combining with the energy method. Applying these theories and methods, we aremore meticulous demonstrates the structure and properties of solutions of the wave equationswith Kirchhoff type in H10() space. It is designed to have a theory instruction function for thesame problem in physics and engineering model. Consequently, this work further enriches anddevelops potential well theory.Firstly, this paper establishes the variational theory corresponding the Kirchhoff’s typewave equations, and defines the equality of energy conservation and also some related function-als. Then the relevant properties of the family of potential wells and invariant sets are givenunder positive energy and non-positive energy. Combining with Galerkin method and energyestimate method, we get the optimum condition for the global existence of the weak solutionscorresponding the above system and then we discuss a long time behavior of the solutions.Moreover under some hypothetical conditions, the global existence is yielded at critical energylevel. Further, this paper studies global nonexistence for such Kirchhoff’s type equations inthree initial energy levels. Under the appropriate assumptions, we study the finite time blow upresults by the potential well method and convex method for the corresponding problem.In summary, the theoretical significance of this paper is reflected in the following innova-tion points: Firstly, in the current many applications and researches with potential well theoryhave focused on the equations with weak damping terms, But this potential wells method willbe applied in nonlinear wave equations with weakly damped terms, employing it we solve theglobal posed-ness of Kirchhoff-type wave equations. Secondly, as far as we know, the study ofKirchhoff-type wave equations at low initial energy are not involved with the asymptotic behav-ior, in this paper we are interested in discussing the open problem, then we obtained asymptotic properties of the solutions of hyperbolic equations of Kirchhoff type with strong and weakdamping terms. Again, this work firstly obtained the global well posed-ness of the solutionsof the Kirchhoff-type wave equations with critical initial energy. Finally, there is an importantbreakthrough at the global nonexistence of the solutions of the Kirchhoff-type wave equations,in finite time blow up result with arbitrary high initial energy E(0)>0is given in this article.