Long Time Behavior Of Solutions For Memory Viscoelastic Evolution Equation With Singularly Oscillating External Forces

In this master academic thesis,we consider the viscoelastic evolution equation with singularly oscillating external forces and memory,and the limiting equation when ε→0 +,where Ω（?）Rⁿ,ε∈(0,1],ρ₀∈[0,1].μ is memory kernel,f is nonlinear function,g is the nonlinear damping,g₀（x,t）+1/（ερ₀）g₁（x,t/ε）stands for singularly oscillating external forces,u_τ（x,t）（the past history of u）is a given number which has to be known for all t≤τ,|ut|^ρ represents the nonlinear material density,and ρ is a real number such that 1<ρ≤2/（n-2）if n≥3;ρ>1,if n=1,2.This master thesis consists of the following parts:In the first part,the research background and current situation of the problem,as well as the main conclusions obtained are described.In the second part,Some concepts and inequalities used in dissertation research are introduced.In the third part,the existence of uniform attractors and upper semicontinuous for memorized viscoelastic problems with singular oscillatory external forces and history memory are studied.First,under appropriate conditions,the well-posedness of the solution is proved,and the existence of uniform（about g^ε∈ H（g^ε））absorbing set of the problem on H₀¹（Ω）×H₀¹（Ω）× L_μ²（R⁺,H₀¹（Ω））for the process associated with the problem is obtained by using a priori estimation;Then the uniform asymptotic compactness of the above problem is proved by using the contraction function method,finally,received the existence of uniform attractor A^ε in space H₀¹（Ω）× H₀¹（Ω）× L_μ²（R⁺,H₀¹（Ω））;In addition,we proved uniform（aboutε）boundedness of the uniform attractor in H₀¹（Ω）×H₀¹（Ω）×L_μ²（R⁺,H₀¹（Ω））.whenε→0⁺ the uniform attractor A^ε of the first equation converges to the uniform attractor of the second equation A⁰.