Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

Ding, Hang; Wang, Renhai; Zhou, Jun

doi:10.1111/sapm.12405

Cited by 12 publications

(3 citation statements)

References 45 publications

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“…Nonlinear damped wave equations appear extensively in physical modeling, such as modeling the transversal vibrations of a homogeneous string and the longitudinal vibrations of a homogeneous bar, subject to viscous effects [10], describing the variation from the configuration at rest of a homogeneous and isotropic linearly viscoelastic solid with short memory [3,12], or as a perturbed wave equations of the Klein-Gordon type appearing in quantum mechanics, etc. Several mathematicians take it as a major concern in recent years, see [1,4,11,14]. In particular, using some tools from critical point theory, Sattinger [13] made a first attempt to deal with nonlinear wave equations when the sign of initial energy is indefinite.…”

Section: Introductionmentioning

confidence: 99%

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

Chen

2023

Mathematische Nachrichten

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We consider the global existence of the solution for nonlinear damped wave equations at arbitrary positive initial energy, that is, . It is well known from a work of G. Filippo and S. Marco [Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006], that which initial data with high energy make the model exists global solutions is still an interesting question. Here, we give it an affirmative answer. For this purpose, we suggest a unified approach, which is also valid where various forms of damping are present. Applying it, we show a sufficient condition to get the global existence of the solution for problem nonlinear damped wave equations.

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Section: Introductionmentioning

confidence: 99%

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

Chen

2023

Mathematische Nachrichten

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“…For the models ( 5) and ( 6), we have proved that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy in our recent paper. 63 However, for problem (1), due to the effects of the strong damping −Δ𝑢 𝑡 and the quasilinear term div(∇𝑢∕ √ 1 + |∇𝑢| 2 ), we must develop some new methods to prove the existence of infinite-time blow-up solutions with subcritical, critical, and arbitrary high initial energy.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity

Ding

Zhou

2022

Stud Appl Math

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This paper investigates the well‐posedness of solutions for the following quasilinear wave equation with strong damping and logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary: utt−normalΔu−div(∇u1+false|∇ufalse|2)−normalΔut=false|ufalse|r−2ulog|u|$u_{tt}-\Delta u-{\rm div}(\frac{\nabla u}{\sqrt {1+|\nabla u|^2}})-\Delta u_t=|u|^{r-2}u\log |u|$, where r≥2$r\ge 2$. By virtue of the classical Faedo–Galerkin method and some technical efforts, we first establish the local well‐posedness of solutions. Then we discuss the dynamical behaviors of solutions in detail: 1.When r≥2$r\ge 2$ and Ifalse(u0false)>0$I(u_0)>0$, we show that the solutions exist globally with subcritical and critical initial energy, where Ifalse(u0false)$I(u_0)$ denotes the Nehari functional with the initial value u0. Especially, under further suitable assumptions about the initial data, we show that the energy functional decays exponentially. 2.When r>2$r>2$ and Ifalse(u0false)<0$I(u_0)<0$, we show that the solutions blow up in finite time with subcritical and critical initial energy. Moreover, by removing the restriction Ifalse(u0false)<0$I(u_0)<0$, we prove that the solutions may blow up in finite time with arbitrary high initial energy. In particular, we derive the upper and lower bounds of the blow‐up time. 3.When r=2$r=2$ and Ifalse(u0false)<0$I(u_0)<0$, we show that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy.

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“…It is worth mentioning that the potential well method was introduced by Sattinger in [31] to study the global existence of solutions to the nonlinear hyperbolic equations. From then on, many researchers applied this method to study the nonlinear evolution equations, see [5,9,10,18,21,22,23,24,25,29,38,43]. Especially, Payne and Sattinger [29] investigated the existence and finite time blow-up of solutions to the initial boundary value problem of semilinear parabolic equations and semilinear hyperbolic equations.…”

Section: Introductionmentioning

confidence: 99%

Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model

Ding¹,

Zhou²

2022

ejde

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This article concerns a Kirchhoff-type parabolic problem on a geodesic ball of hyperbolic space. Firstly, we obtain conditions for finite time blow-up, and for the existence of global solutions for $J(u_0)\leq d$, where $J(u_0)$ denotes the initial energy and d denotes the depth of the potential well. Secondly, we estimate the upper and lower bounds of the blow-up time. In addition, we derive the growth rate of the blow-up solution and the decay rate of the global solution. Thirdly, we establish a new finite time blow-up condition which is independent of d and prove that the solution can blow up in finite time with arbitrary high initial energy, by using this blow-up condition. Finally, we present some equivalent conditions for the solution existing globally or blowing up in finite time.

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Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

Cited by 12 publications

References 45 publications

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity

Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model

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