一类带积分项粘弹性梁方程的定解问题

THE PROBLEM OF DETERMINING SOLUTION FOR A CLASS OF VISCOELASTIC BEAM INTEGRO DIFFERENTIAL EQUATIONS

讨论下列初边值问题ｕｔｔ＋ｕｘｘｘｘ－∫ｔ０λ（ｔ－τ）ｕｘｘｘｘ（ｘ，τ）ｄτ＝ｇ（ｕ）＋ｈ（ｕｔ），０＜ｘ＜Ｌ，ｔ＞０ｕ｜ｘ＝０＝ｕ｜ｘ＝Ｌ＝０，ｔ≥０ｕｘｘ｜ｘ＝０＝ｕｘｘ｜ｘ＝Ｌ＝０，ｔ≥０ｕ｜ｔ＝０＝φ（ｘ），ｕｔ｜ｔ＝０＝ψ（ｘ），０≤ｘ≤Ｌ在一定条件下，证明了这个问题整体经典解的存在性、唯一性和稳定性。

The global classical solutions of the following problem were studied u tt +u xxxx -∫ t 0λ(t-τ)u xxxx (x,τ) d τ=g(u)+h(u t),0<x<L,t>0 〖HJ*4]u| x=0 =u| x=L =0,t≥0 u xx | x=0 =u xx | x=L =0,t≥0 u| t=0 =φ(x),u t| t=0 =ψ(x),0≤x≤L under certain conditions on g,h and λ,the unique existence and stability of the global classical solution of this problem could be obtained.