Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

Download Full-text

Blow Up of Solutions for a System of Nonlinear Viscoelastic Equations with Damping Terms in Rn

Zeitschrift für Naturforschung A ◽

10.1515/zna-2009-3-404 ◽

2009 ◽

Vol 64 (3-4) ◽

pp. 180-184

Author(s):

Wenjun Liu ◽

Shengqi Yub

Keyword(s):

Initial Data ◽

Differential Inequality ◽

Blow Up ◽

Coupled System ◽

Initial Energy ◽

Relaxation Functions ◽

Nonlinear Viscoelastic ◽

Linear Damping ◽

Viscoelastic Equations ◽

Damping And Source Terms

Abstract We consider a coupled system of nonlinear viscoelastic equations with linear damping and source terms. Under suitable conditions of the initial data and the relaxation functions, we prove a finitetime blow-up result with vanishing initial energy by using the modified energy method and a crucial lemma on differential inequality

Download Full-text

Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

Journal of Function Spaces ◽

10.1155/2021/5573959 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Kh. Zennir ◽

H. Dridi ◽

S. Alodhaibi ◽

S. Alkhalaf

Keyword(s):

Boundary Value Problem ◽

Blow Up ◽

Modern Science ◽

Global Solutions ◽

Coupled System ◽

Evolution System ◽

Variable Exponents ◽

Scientific Interest ◽

Time Blowup ◽

Pseudoparabolic Equations

The most important behavior for evolution system is the blow-up phenomena because of its wide applications in modern science. The article discusses the finite time blowup that arise under an appropriate conditions. The nonsolvability of boundary value problem for damped pseudoparabolic differential equations with variable exponents is investigated. Such problem has been previously studied in the case if p and q are constants. New here is the case of variables of nonlinearity p and q which make the problem has a scientific interest.

Download Full-text

A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy

Boundary Value Problems ◽

10.1186/s13661-021-01538-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Qian Li

Keyword(s):

Initial Data ◽

Finite Time ◽

Wave Equations ◽

Blow Up ◽

Sufficient Conditions ◽

Auxiliary Function ◽

Initial Energy ◽

Positive Initial Energy ◽

Viscoelastic Wave ◽

Viscoelastic Equations

AbstractThis article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

Download Full-text

Blow-Up and Global Existence of Solutions for the Time Fractional Reaction–Diffusion Equation

Mathematics ◽

10.3390/math9243248 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3248

Author(s):

Linfei Shi ◽

Wenguang Cheng ◽

Jinjin Mao ◽

Tianzhou Xu

Keyword(s):

Initial Data ◽

Diffusion Equation ◽

Blow Up ◽

Sufficient Conditions ◽

Global Solutions ◽

Reaction Diffusion ◽

Reaction Diffusion Equation ◽

Time Behavior ◽

Global Existence Of Solutions ◽

Long Time

In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed.

Download Full-text

Global solutions and blowing-up solutions for a nonautonomous and nonlocal in space reaction-diffusion system with Dirichlet boundary conditions

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0054 ◽

2020 ◽

Vol 23 (4) ◽

pp. 1025-1053

Author(s):

Marcos J. Ceballos-Lira ◽

Aroldo Pérez

Keyword(s):

Boundary Conditions ◽

Dirichlet Boundary ◽

Blow Up ◽

Sufficient Conditions ◽

Global Solutions ◽

Coupled System ◽

Dirichlet Boundary Conditions ◽

Weakly Coupled System ◽

Blowing Up ◽

Fractional Diffusions

AbstractWe give sufficient conditions for global existence and finite time blow up of positive solutions for a nonautonomous weakly coupled system with distinct fractional diffusions and Dirichlet boundary conditions. Our approach is based on the intrinsic ultracontractivity property of the semigroups associated to distinct fractional diffusions and the study of blow up of a particular system of nonautonomus delay differential equations.

Download Full-text

PARAXIAL APPROXIMATION OF THE VLASOV-MAXWELL SYSTEM: LAMINAR BEAMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202594000121 ◽

1994 ◽

Vol 04 (02) ◽

pp. 203-221 ◽

Cited By ~ 3

Author(s):

A. NOURI

Keyword(s):

Electromagnetic Field ◽

Charged Particle ◽

Initial Data ◽

Global Existence ◽

Local Existence ◽

Sufficient Conditions ◽

External Electromagnetic Field ◽

Evolution System ◽

Lagrangian Coordinates ◽

Maxwell System

The Vlasov-Maxwell stationary system for charged particle laminar beams is studied with a paraxial model of approximation. It leads to a degenerate evolution system, which local existence is proved. Then, using lagrangian coordinates, with sufficient conditions on the initial data and the external electromagnetic field, it is shown that global existence is possible.

Download Full-text

Global solutions to a hyperbolic-parabolic coupled system with large initial data

Acta Mathematica Scientia ◽

10.1016/s0252-9602(09)60059-x ◽

2009 ◽

Vol 29 (3) ◽

pp. 629-641 ◽

Cited By ~ 14

Author(s):

Guo Jun ◽

Xiao Jixiong ◽

Zhao Huijiang ◽

Zhu Changjiang

Keyword(s):

Initial Data ◽

Global Solutions ◽

Coupled System ◽

Large Initial Data

Download Full-text

Flow invariance for perturbed nonlinear evolution equations

Abstract and Applied Analysis ◽

10.1155/s1085337596000231 ◽

1996 ◽

Vol 1 (4) ◽

pp. 417-433 ◽

Cited By ~ 22

Author(s):

Dieter Bothe

Keyword(s):

Evolution Equations ◽

Mild Solutions ◽

Sufficient Conditions ◽

Global Solutions ◽

Reaction Diffusion ◽

Accretive Operator ◽

Nonlinear Evolution Equations ◽

Evolution System ◽

Reaction Diffusion Systems ◽

Closed Sets

LetXbe a real Banach space,J=[0,a]⊂R,A:D(A)⊂X→2X\ϕanm-accretive operator andf:J×X→Xcontinuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed setsK⊂Xfor the evolution systemu′+Au∍f(t,u) on J=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraintsu(t)∈K(t)onJ. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of typeut=ΔΦ(u)+g(u) in (0,∞)×Ω, Φ(u(t,⋅))|∂Ω=0, u(0,⋅)=u0under certain assumptions on the setΩ⊂Rnthe functionΦ(u1,…,um)=(φ1(u1),…,φm(um))andg:R+m→Rm.

Download Full-text

Nonexistence of global solutions for a class of viscoelastic wave equations

Discrete and Continuous Dynamical Systems - Series S ◽

10.3934/dcdss.2021134 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Jorge A. Esquivel-Avila

Keyword(s):

Wave Equations ◽

Evolution Equations ◽

Blow Up ◽

Global Solutions ◽

Initial Energy ◽

Nonlinear Evolution Equations ◽

Material Behavior ◽

Memory Term ◽

Viscoelastic Wave ◽

Deformable Bodies

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

Download Full-text

Finite time blow-up of complex solutions of the conserved Kuramoto–Sivashinsky equation in ℝd and in the torus 𝕋d, d ⩾ 1

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.54 ◽

2019 ◽

Vol 149 (5) ◽

pp. 1175-1188

Author(s):

Léo Agélas

Keyword(s):

Sobolev Space ◽

Initial Data ◽

Besov Space ◽

Finite Time ◽

Blow Up ◽

Global Solutions ◽

Large Initial Data ◽

Complex Solutions ◽

Complex Valued ◽

Sivashinsky Equation

AbstractWe consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

Download Full-text