Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term
The following Cauchy problem for the dissipative Kirchhoff equation is considered: \[ u^{\prime\prime}+\delta u^{\prime}+m(\mid A^{\frac{1}{2}}u\mid^2_H)Au=0, \] \[ u(0)=u_0,\,u^{\prime}(0)=u_1, \] where \(H\) is a Hilbert space, \(\delta>0\), \(A\) is a self-adjoint nonnegative operator in \(H\) and \(m(r)\) is a nonnegative function such that \(m(0)=0,\,m(\mid A^{\frac{1}{2}}u_0\mid^2_H)>0.\) For small initial data the existence of global solutions as well as decay of the energy as \(t\to\infty\) are proved. Earlier, see [\textit{A. T. Cousin, C. L. Frota, N. A. Lar'kin}, Differ. Integral Equ. 15, No. 10, 1219--1236 (2002; Zbl 1011.35097)], similar results were obtained for the equation \[ u_{tt}-\phi(x,t)M\left(\int_{\omega} \mid \triangledown u(s,t)\mid^2ds\right)\Delta u +g(x,t,u_t)=0. \]
scientific article; zbMATH DE number 5077857
scientific article