Well-posedness Of Solutions For Several Kirchhoff Type Evolution Equations

In this dissertation,the well posedness of solutions for several Kirchhoff type evolution equations,including the global existence and finite time blow up of solutions,are studied.It is of great significance for the theoretical study and practical application of nonlinear evolution equations.Specifically,this dissertation consists of the following five parts.In the first part,the research background and significance of Kirchhoff-type partial differential equation and its well posedness are introduced.Then we give the specific problems in the present dissertation and the rest of the structure.In the second part,the initial boundary value problem for a Kirchhoff-type parabolic equation with logarithmic nonlinearity is studied.By employing the potential well theory and some differential inequality techniques,a new blow-up condition,the upper bound of the blow-up time,and the lower bound of the growth rate of blow-up solutions are obtained.Thus,the main result improves a recent blow-up result and makes it more applicable.In the third part,the initial boundary value problem for a Kirchhoff-type hyperbolic equation with generalized nonlinearity is considered,where the nonlinearity satisfies some assumptions.First,a blow up condition independent of the depth of the potential well is found and the upper bound of the blow up time is given.Furthermore,the finite time blow up result of solutions with arbitrary high initial energy is obtained by using this blow up condition.It generalizes the previous blow up result and makes the conclusion of this equation more abundant.In the fourth part,the initial boundary value problem for a Kirchhoff-type hyperbolic equation with logarithmic nonlinearity is introduced.Firstly,by employing the classical Galerkin method and potential well method,we show the existence of the global weak solution with subcritical and critical initial energy.Secondly,we obtain the finite time blow up results with subcritical and critical initial energy by using the concavity method.In addition,under some suitable conditions,we also estimate the upper and lower bounds of the blow-up time for the blow-up solution by using the differential inequality technique.Finally,we establish a new finite time blow up condition which is independent of the depth of the potential well and we obtain the upper bound of the blow-up time for the blow-up solution.Furthermore,we also prove the weak solution can blow up in finite time at arbitrary initial energy level by using this condition.It is understood that the well posedness of solutions for the problem is studied for the first time with a certain degree of innovation.In the fifth part,the above research works,including the main results completed and the research plan in next step,are summarized and prospected.