The Blow-up And Perturbation Of Solution Of Initial-boundary Problem For Nonlinear Evolution Equation

This paper includes three sections:In sectionâ… , we give the background and the main theorem.In sectionâ…¡, we consider the following wave equations:where,Ω(?) Rⁿ denotes a bounded domain with smooth boundary (?)Ω,(?), a(x)∈C₀¹(Ω),a(x)>a₀>0 and a₀ is a positive constant. Denotes Q_T=[0,T]×(?)Ω,T>0. First we discuss the global existence and uniqueness of solution, and then prove the Blow-up of solution under the appropriate conditions.In sectionâ…¢, we consider the perturbed solution of the following parabolic equation:where (?),is a small positive parameter,σ, T is positive constant,Ω_ε={(r,φ)|0≤r < r₀(φ,ε), 0≤φ≤2Ï€} denotes a bounded convex domain in Rⁿ, and (?)Ω_ε:r=r₀(φ,ε) signifies the smooth boundary ofΩ_εof class C^1+α (α∈(0,1) is a H(?)lder exponent), L is a uniformly elliptic operator, that is, exist positive constantλ,Λto meet(?), and (?) is abounded constant inΩ_ε, (?) only depend on r .(?) denotes the outward derivative on (?)Ω_ε,K₁(x,y) and K₂(x,y) are integral nuclear onΩ_εand (?)Ω_εrespectively.