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Initial value problems for a class of nonlinear integro-partial differential equations

Authors:

Guo Ai

Affiliation:

Gansu University of Technology, Lanzhou 730050, P. R. China

Abstract:

In this paper, the problem of global existence of solutions to the following initial value problem is studied:

$left{ begin{gathered} u_{tt} - au_{xxt} - p(u_x )_x - int_0^t {lambda (t - s)q(u_x )_x ds} = f(x,t) hfill ( - infty< x< + infty ), t > 0, hfill u left| {_{t = 0} = varphi (x), u_t } right|_{t = 0} = psi (x) ( - infty< x< + infty ) hfill end{gathered} right.$

, which comes from viscoelastic mechanics. By making use of integral estimates method, it is proved that this problem has a global solution if, in addition to certain regularity assumptions on the given functions, the following conditions are satisfied: p′(s) ≥ c₁ > 0, |q′(s)| ≤ const, λ(0) < 0, λ′(0) < λ²(0)

Keywords:

initial value problem integro-partial differential equation classical solution

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