Global nonexistence of solutions for viscoelastic wave equations of Kirchhoff type with high energy
Summary: We consider viscoelastic wave equations of the Kirchhoff type \[ u_{tt} - M(||\nabla u||^2_2) \Delta u + \int^t_0 g(t - s) \Delta u(s)ds + u_t = |u|^{p-1}u \] with Dirichlet boundary conditions, where \(||\cdot||_p\) denotes the norm in the Lebesgue space \(L^p\). Under some suitable assumptions on \(g\) and the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense that \[ \lim_{t \rightarrow T^{\ast -}} (||u(t)||^2_2 + \int^t_0 ||u(s)||^2_2 ds) = \infty \] for some \(0 < T^\ast < +\infty\).
scientific article; zbMATH DE number 6032365
scientific article