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

Authors:

Guo Ai

Institution:

(1) Gansu University of Technology, 730050 Lanzhou, 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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