Lower And Upper Bounds For The Blow-up Time In Viscoelastic Hyperbolic Equations With Variable Sources And Damping Terms

This paper deals with the following problemBy constructing a new control function,the relationship between the term(?)|u|p(x,t)dx and the initial energy is established,and then we prove that the solution of the above problem blows up in a finite time for a positive initial energy,especially,it is only assumed that pr(x,t)is integrable rather than uniformly bounded.This weak condition is rarely seen in the case of variable exponents.Furthermore,a lower bound for the blow-up time is established.The main conclusions are as follows:Theorem 1.Suppose that the exponent p(x,t),m(x,t)satisfies 1<p-≤ p(x,t)≤p+<∞,1<m-≤m(x,t)≤ m+<∞,and the following conditions holdThen Problem(0.1)has a unique local solution u ∈C([0,T);H01(Ω))∩ Lp-(O,T;Lp(x,t)(Ω)),ut ∈C([0,T);L2(Ω))∩ Lm(Ω ×(0,T)),for some T.Theorem 2.Assume that(H2)and(H3)of Theorem 1 hold,and that the following conditions are satisfied:(H6)there exists a sufficiently small 0<ε0<1 such that 1-ε0≤k<1.Then the blow-up time T*satisfies the following estimate where the coefficients are defined by where B is the embedding constant with H01(Ω)→ Lp+(Ω).