Research On The Blow-up Of The Solution For Nonlinear Wave Equation With Acoustic Boundary Conditions

The initial-boundary value problem of partial differential equations is a research hotspot in mathematics,which is closely related to various natural phenomena and practical appli-cations.This paper considers the nonlinear divergence-form wave equation with initial conditions and acoustic boundary conditions under the interaction between interior/boundary damping terms and source terms.The main innovations of this paper are as follows:(a)Compared with the previous literatures,the problem studied in this paper is more general.(b)The assumptions for damping terms and source terms are weaker.That is,we assume that damping terms have no restrictions to polynomial growth near zero,and source terms could be general nonlinear functions which are controlled by polynomial growth.(c)We answer an open problem proposed in[19]and generalize the blow-up result for E(0)<E1.In order to overcome these difficulties,based on the so-called Georgiev-Todorova method,we utilize the technique of partition of the boundary to obtain some more delicate estimates.Finally,under some suitable assumptions,we prove that the solution of this initial-boundary value problem blows up at finite time and give the upper bound of the blow-up time Tmax.In the meantime,we find that inequality is an important tool to solve the problems related to partial differential equations.Under the circumstance,this paper refines and generalizes Hardy's inequality in Sobolev space.This paper refines the proof of Hardy's inequality in[12,22],and extends the Hardy's inequality in Sobolev space from two aspects.That is to say,we extend the integral estimation function from u/|x| to u/|x|? with suitable ?>0,and extend the space dimension from n? 3 to ? 2.The Hardy's inequality in[12,22]is the special case of our results.