Abstract:

In this paper, we study the problem of global existence of solutions to the following initial-boundary value problem

       u_tt-au_xxt-p(u_x)-∫^t₀λ(t-s)q(u_x)_xds = f(x,t),00

       u|_x=0 = 0,u|_x=1 = 0,t≥0,

       u|_t=0 =φ(x),u_t|_t=0 =ψ(x), 0≤x≤1,

which comes from viscoelastic mechanics. By making use of Galerkin's approximation method, we prove that this problem has a unique global solution if, in addition to certain regularity assumptions on the given functions, the following conditions are satisfied

    p′(s)≥0, |p′(s)-q′(s)|≤const., λ(0)<0, λ′(0)<λ²(0).

Key words: Ｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ, Ｉｎｔｅｇｒｏｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ, Ｃｌａｓｓｉｃａｌｓｏｌｕｔｉｏｎ．