| In this paper,a class of hyperbolic equations with strong damping and logarithmic source terms are studied.Consider the following initial boundary value problem:For this problem,we first prove the local existence of the weak solution by using Galerkin method and the contraction mapping principle.Furthermore,in the potential well framework,combined with convex method,the blow up results at finite time are obtained.In addition,the upper and lower bounds of the blow up time are derived.The chapter arrangement of this paper is as follows:The first chapter is the introduction,it mainly introduces the research object,the background of evolution equation with logarithmic term and evolution equation with strong damping term.The second chapter is the preliminary knowledge,which gives the inequalities commonly used in this paper,the structure of potential well and some key lemmas.In chapter 3,the existence and uniqueness theorem of local solutions is given and proved.In the fourth chapter,the theorem and proof of blow up in finite time are obtained.The main conclusions of this paper are as follows:Conclusion 1:Suppose that u0∈H,u1(x)∈L2(Ω).Then there exist a T>0 such that the above problem admits a unique weak solution u(x,t)on Ω×[0,T]satisfying u∈C([0,T],H)∩C1([0,T],L2(Ω))∩C2([0,T],H-2(Ω)).Conclusion 2:Assume u0∈H\{0},u1∈L2(Ω)satisfy E(0)<d,I(u0)<0,then the solution u(x,t)of the above problem blows up in finite time,that is,#12 Moreover,the blow up time T*satisfies the following estimate#12Conclusion 3:On the basis of conclusion 2,and 2<p<1+n/(n-4),then the blow up time T*satisfies the following estimate#12... |
