Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces

Abstract

In this article, we deal with the initial boundary value problem for a viscoelastic system related to the quasilinear parabolic equation with nonlinear boundary source term on a manifold $\mathbb{M}$ with corner singularities. We prove that, under certain conditions on relaxation function $g$, any solution $u$ in the corner-Sobolev space $\mathcal{H}^{1,(\frac{N-1}{2},\frac{N}{2})}_{\partial^{0}\mathbb{M}}(\mathbb{M})$ blows up in finite time. The estimates of the life-span of solutions are also given.