scholarly journals Blow-Up of Solutions for a Class Quasilinear Wave Equation with Nonlinearity Variable Exponents

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zakia Tebba ◽  
Hakima Degaichia ◽  
Mohamed Abdalla ◽  
Bahri Belkacem Cherif ◽  
Ibrahim Mekawy
Keyword(s):  
Wave Equation ◽  
Finite Time ◽  
Variable Exponent ◽  
Blow Up ◽  
Initial Energy ◽  
Positive Energy ◽  
Variable Exponents ◽  
Quasilinear Wave Equation ◽  
New Class ◽  
Dispersion Term

This work deals with the blow-up of solutions for a new class of quasilinear wave equation with variable exponent nonlinearities. To clarify more, we prove in the presence of dispersion term − Δ u t t a finite-time blow-up result for the solutions with negative initial energy and also for certain solutions with positive energy. Our results are extension of the recent work (Appl Anal. 2017; 96(9): 1509-1515).

scholarly journals Blow-Up of Certain Solutions to Nonlinear Wave Equations in the Kirchhoff-Type Equation with Variable Exponents and Positive Initial Energy

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Loay Alkhalifa ◽  
Hanni Dridi ◽  
Khaled Zennir
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Blow Up ◽  
Type Equation ◽  
Initial Energy ◽  
Nonlinear Wave Equations ◽  
Variable Exponents ◽  
Quasilinear Wave Equation ◽  
Positive Initial Energy

This paper is concerned with the blow-up of certain solutions with positive initial energy to the following quasilinear wave equation: u t t − M N u t Δ p · u + g u t = f u . This work generalizes the blow-up result of solutions with negative initial energy.

scholarly journals General decay and blow up of solutions for a variable-exponent viscoelastic double-Kirchhoff type wave equation with nonlocal degenerate damping

2021 ◽  
Author(s):  
Mohammad Shahrouzi ◽  
Jorge Ferreira ◽  
Erhan Pişkin
Keyword(s):  
Wave Equation ◽  
Variable Exponent ◽  
Blow Up ◽  
Type Equation ◽  
Initial Energy ◽  
General Decay ◽  
Kirchhoff Type ◽  
Type Wave ◽  
Positive Initial Energy ◽  
Viscoelastic Term

In this paper we consider a viscoelastic double-Kirchhoff type wave equation of the form $$ u_{tt}-M_{1}(\|\nabla u\|^{2})\Delta u-M_{2}(\|\nabla u\|_{p(x)})\Delta_{p(x)}u+(g\ast\Delta u)(x,t)+\sigma(\|\nabla u\|^{2})h(u_{t})=\phi(u), $$ where the functions $M_{1},M_{2}$ and $\sigma, \phi$ are real valued functions and $(g\ast\nabla u)(x,t)$ is the viscoelastic term which are introduced later. Under appropriate conditions for the data and exponents, the general decay result and blow-up of solutions are proved with positive initial energy. This study extends and improves the previous results in the literature to viscoelastic double-Kirchhoff type equation with degenerate nonlocal damping and variable-exponent nonlinearities.

Well-posedness and blow-up properties for the generalized Hartree equation

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

Blow up of Solution for the Generalized Boussinesq Equation with Damping Term

2006 ◽  
Vol 61 (5-6) ◽  
pp. 235-238
Author(s):  
Necat Polat ◽  
Doğan Kaya
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Damping Term ◽  
Initial Boundary Value ◽  
Generalized Boussinesq Equation

We consider the blow up of solution to the initial boundary value problem for the generalized Boussinesq equation with damping term. Under some assumptions we prove that the solution with negative initial energy blows up in finite time

Global solutions and finite time blow-up for fourth order nonlinear damped wave equation

2018 ◽  
Vol 59 (6) ◽  
pp. 061503 ◽  
Author(s):  
Runzhang Xu ◽  
Xingchang Wang ◽  
Yanbing Yang ◽  
Shaohua Chen
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Damped Wave Equation ◽  
Nonlinear Damped Wave Equation
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