Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

TL;DR

This paper analyzes the blow-up behavior of solutions to a hyperbolic PDE modeling wave propagation in viscoelastic media with supercritical nonlinearities, linear memory, and damping, identifying conditions under which solutions become unbounded.

Contribution

It provides new blow-up results for a viscoelastic wave equation with supercritical source and infinite memory, extending previous well-posedness studies.

Findings

Solutions blow up when the nonlinear source dominates damping and memory effects.

Blow-up occurs under negative total energy or large positive quadratic energy.

The model features a full past history memory, unlike typical finite-memory models.

Abstract

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: \begin{align*} \begin{cases} u_{tt}- k(0) \Delta u - \int_0^{\infty} k'(s) \Delta u(t-s) ds + |u_t|^{m-1}u_t=|u|^{p-1}u, \;\;\;\;\; \Omega \times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } \Omega \times (-\infty,0], \end{cases} \end{align*} where $\Omega$ is a bounded domain in $\mathbb R^3$ with a Dirichl\'et boundary condition. The relaxation kernel $k$ is monotone decreasing and $k(\infty)=1$. We study blow-up of solutions when the source is stronger than dissipations, i.e., $p> \max\{m,\sqrt{k(0)}\}$, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to $-\infty$, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.