Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources

We consider, in this paper, the following nonlinear equation with variable exponents:where a, b > 0 are constants and the exponents of nonlinearity m, p, and r are given functions. We prove a finite-time blow-up result for the solutions with negative initial energy and for certain solutions with positive energy. KEYWORDS blowup, nonlinear damping, Sobolev spaces with variable exponentsA > 0, 0 < < 1. The term Δ r(·) u = div(|∇u| r(·)−2 ∇u) is called r(·)− Laplacian.

show abstract

“…This work is considered to be an improvement for that of Antontsev. 24 In the work of Sun et al, 27 they looked into the following equation:…”

Section: )mentioning

confidence: 99%

Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

Messaoudi

Talahmeh

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…. For more works, the interested readers may refer to [23,24]. When p is a function, the authors in [2,20] applied Kaplan's method to establish the nonglobal existence and global existence results for Problem (1) in the absence of the viscoelastic term and the damping term.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…The proof of the existence of solutions relays on Galerkin approximation technique and the contraction mapping theorem. For more details, we may refer to [7,13,23]. Set…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Lower and upper bounds for lifespan of solutions to viscoelastic hyperbolic equations with variable sources and damping terms

Dai

Zhang

2019

J Inequal Appl

View full text Add to dashboard Cite

The aim of this paper is to study bounds for lifespan of solutions to the following equation: u tt-u + t 0 g(t-τ) u(τ) dτ + |u t | m(x,t)-2 u t = |u| p(x,t)-2 u under homogeneous Dirichlet boundary conditions. It is worth pointing out that it is not a trivial generalization for constant-exponent problems because we have to face some essential difficulties in studying such problems. The first difficulty is that the monotonicity of the energy functional fails. Another one is that there exists a gap between the norm and the modular to the generalized function space, which leads to the failure of the Poincaré inequality for modular form. To overcome such difficulties, the authors construct control function and apply new energy estimates to establish the quantitative relationship between the source Ω |u| p(x,t) dx and the initial energy, and then obtain the finite-time blow-up of solutions for a positive initial energy, especially, the authors only assume that p t (x, t) is integrable rather than uniformly bounded. Such weak conditions are seldom seen for the variable exponent case. At last, an estimate of lower bound for lifespan is established by applying differential inequality argument and energy inequalities.

show abstract

“…This work is considered to be an improvement for that of [6]. In [82], Sun et al looked into the following equation…”

Section: Blowup In the Case Of Variable-exponent Nonlinearitiesmentioning

confidence: 99%

On wave equation: review and recent results

Messaoudi

Talahmeh

2017

Arab. J. Math.

View full text Add to dashboard Cite

The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. Mathematics Subject Classification

show abstract

Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources

Cited by 36 publications

References 19 publications

Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

Blowup in solutions of a quasilinear wave equation with variable‐exponent nonlinearities

Lower and upper bounds for lifespan of solutions to viscoelastic hyperbolic equations with variable sources and damping terms

On wave equation: review and recent results

Contact Info

Product

Resources

About