Blow Up Of Arbitrarily Positive Initial Energy Solutions For A Quasilinear Wave Equation With Variable Exponent Nonlinearities

In this paper,we mainly discuss the following problems about homogeneous Dirichlet boundary conditions.（?）where a,b are positive constants,Ω is a bounded region with smooth boundary in Rⁿ.m（·）,p（·）,r（·）are measurable functions on Ω.Satisfy 2≤min{m^-,r^-}≤max{m⁺,r⁺}<p^-≤p（x）<p⁺≤r_*（x）,（?） and （?）We suppose that m（·）,p（·）,r（·）satisfies the following conditions |p（x）-p（y）|≤（-A）/（log|x-y|）,x,y∈Ω,|x-y|<δ,（3） here A>0,0<δ<1.m（·）,p（·）,r（·）are measurable functions on Ω,more extensive application scenarios and effects.In this paper,by applying the basic properties of variable index space,embedding theorem and Gronwall inequality,we prove that for any positive initial energy,solutions blow up in finite time.The main conclusions are as follows:proposition 1.Suppose u0∈W01,R（·）（Ω）,u1∈L2（Ω）,m（·）,p（·）,r（·）satisfy（2）and（3）,the problem（1）has a weak solution satisfied u ∈ L^∞(（0,T）,W₀^1,r（·）（Ω）),u_t∈L^∞（（0,T）,L²（Ω））,U_tt ∈ L^∞(（0,T）,W₀^{-1,r’（·）}（Ω）),（1/r（·））+（1/r’（·））=1.Theorem 1.Suppose m（x）,r（x）,p（x）satisfy（2）,λ1 is the first eigenvalue of-△,u（t）is the solution of equation（1）,satisfying （?）,then u（t）blow up in a finite time.The existence of sufficiently small ε makes the following conditions valid（?）.