A blow-up result for Kirchhoff-type equations with high energy

Abstract

In this paper, we consider the following Kirchhoff type equation:

Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

“…Motivated by the above research, we consider problem (1.1) for m = 1 in this paper and establish a global nonexistence result for certain solutions with arbitrarily high energy. In this way, we can extend the result of [27] to nonzero term g and the result of [22] to nonconstant M (s). We also obtain the new result for blow-up properties of local solution with arbitrarily high energy.…”

“…By applying the potential well method he obtained the blow-up properties with positive initial energy E(0). Recently, Zeng et al [27] studied equation (1.2) for the case g(u t ) = u t with initial condition and zero Dirichlet boundary condition. By using the concavity argument, they proved that the solutions to equation (1.2) blow up in finite time with arbitrarily high energy.…”

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

“…We may refer the tools to deal with (1)- (5) with viscoelastic terms. By the well-known works [40][41][42][43][44], we see that the supercritical case (0) > is not easy to deal with. Filippo and Marco [45] made the initial attempt to consider the global wellposedness of hyperbolic problem at high initial energy level − Δ − Δ + = | | −2 .…”

Non-existence of global solutions to nonlinear wave equations with positive initial energy

“…The concavity method and its modifications are employed to find sufficient conditions of blow up of solutions with arbitrary large positive initial energy to the Cauchy problem and initial boundary value problems for nonlinear Klein -Gordon equation, damped Kirchhoff-type equation, generalized Boussinesq equaton, quasilinear strongly damped wave equations and some other equations (see, e.g. [1,4], [16]- [18], [25,26,35,36] and references therein).…”

“…where f : R → R is a continuous function that satisfies the condition (35). For local solvability of initial boundary value problem for equations (36) and (37) we refer to [6], where the authors employed the fact that the semigroups generated by corresponding linear problems are analytic (see [9]). • Initial boundary value problem for quasilinear strongly damped wave equation of the form…”