Blow up property for viscoelastic evolution equations on manifolds with conical degeneration

TL;DR

This paper investigates the blow-up behavior of solutions to a nonlinear viscoelastic evolution equation on manifolds with conical degeneration, establishing finite-time blow-up conditions and bounds on blow-up time.

Contribution

It proves finite-time blow-up for solutions with certain parameters and initial conditions on manifolds with conical singularities, and provides lower bounds for blow-up time.

Findings

Solutions blow up in finite time for p > m and positive initial energy.

Constructed lower bounds for blow-up time under specific data assumptions.

Established existence of blow-up solutions in cone Sobolev spaces.

Abstract

This paper is concerned with the study of the nonlinear viscoelastic evolution equation with strong damping and source terms, described by \[u_{tt} - \Delta_{\mathbb{B}}u + \int_{0}^{t}g(t-\tau)\Delta_{\mathbb{B}}u(\tau)d\tau + f(x)u_{t}|u_{t}|^{m-2} = h(x)|u|^{p-2}u , \hspace{1 cm} x\in int\mathbb{B}, t > 0,\] where $\mathbb{B}$ is a stretched manifold. First, we prove the solutions of problem {1.1} in cone Sobolev space $\mathcal{H}^{1,\frac{n}{2}}_{2,0}(\mathbb{B}),$ admit a blow up in finite time for $p > m$ and positive initial energy. Then, we construct a lower bound for obtained blow up time under appropriate assumptions on data.