Non-linear Visco-elastic Wave Equation, Initial Boundary Value Problem Solution Blasting And Attenuation

In this paper, we mainly study the existence of the solution for initial-boundary value problem of nonlinear viscoelastic wave equation, and show that the local solution blows up within finite time and the global solution is decay. This paper consists of two chapters and five sections.In chapter 1, we prove the existence of the local solution and show that the local solution blows up within finite time.In section 1.1, we give some preliminary knowledge, that is Sobolev imbedding theorem and several kind of inequalities (H(?)lder inequality, Minkowskis's inequality, Young inequality, Poincare inequality, and Gronwall inequality, and so on).In section 1.2, using the contraction mapping principle, we prove the existence of the local solution and a energy inequality, i.e. if u₀∈H₀¹(Ω), u₁∈L²(Ω), and the condition of (H₃) holds, then there exists a weak solution u to problem (P) and u∈W, if T is small enough.The blow-up of the local solution will be given in section 1.3, we show that if the conditions of (H₁) and (H₄) hold, and (u₀,u₁)∈H₀¹(Ω)×L² satisfyand∫₀^t g(s)ds is appropriate small, then the weak solution blows up.In Chapter 2, we give the proof of the decay of the global solution. In orde to prove our conclusion, we give some preliminary knowledge in section 2.1. In section 2.2, we prove that the global solution decays by implying the functional whereε₁,ε₂ are arbitrary constants, and the definitions ofφ(t).X(t) are given by...