scholarly journals A New Result for a Blow-up of Solutions to a Logarithmic Flexible Structure with Second Sound

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ahlem Merah ◽  
Fatiha Mesloub ◽  
Salah Mahmoud Boulaaras ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Heat Flux ◽  
Weak Solution ◽  
Time Delay ◽  
Blow Up ◽  
Initial Energy ◽  
Flexible Structure ◽  
Second Sound ◽  
Infinite Time ◽  
Cattaneo’S Law

This paper is concerned with a problem of a logarithmic nonuniform flexible structure with time delay, where the heat flux is given by Cattaneo’s law. We show that the energy of any weak solution blows up infinite time if the initial energy is negative.

scholarly journals Blowup of solutions with positive energy in nonlinear thermoelasticity with second sound

2004 ◽  
Vol 2004 (3) ◽  
pp. 201-211 ◽  
Author(s):  
Salim A. Messaoudi ◽  
Belkacem Said-Houari
Keyword(s):  
Heat Flux ◽  
Initial Energy ◽  
Positive Energy ◽  
Second Sound ◽  
Fourier’S Law ◽  
Blowup Of Solutions ◽  
Nonlinear Thermoelasticity ◽  
Positive Initial Energy ◽  
Fourier's Law ◽  
Cattaneo’S Law

This work is concerned with a semilinear thermoelastic system, where the heat flux is given by Cattaneo's law instead of the usual Fourier's law. We will improve our earlier result by showing that the blowup can be obtained for solutions with “relatively” positive initial energy. Our technique of proof is based on a method used by Vitillaro with the necessary modifications imposed by the nature of our problem.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Le Thi Phuong Ngoc ◽  
Khong Thi Thao Uyen ◽  
Nguyen Huu Nhan ◽  
Nguyen Thanh Long
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Local Existence ◽  
Initial Energy ◽  
Dirichlet Boundary Conditions ◽  
Local Existence And Uniqueness ◽  
Pseudoparabolic Equations

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>

scholarly journals Blowup for nonlinearly damped viscoelastic equations with logarithmic source and delay terms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Energy Method ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Initial Energy ◽  
Damping Term ◽  
Finite Point ◽  
Source Effect ◽  
Positive Initial Energy ◽  
Viscoelastic Equations

AbstractIn this work, we investigate blowup phenomena for nonlinearly damped viscoelastic equations with logarithmic source effect and time delay in the velocity. Owing to the nonlinear damping term instead of strong or linear dissipation, we cannot apply the concavity method introduced by Levine. Thus, utilizing the energy method, we show that the solutions with not only non-positive initial energy but also some positive initial energy blow up at a finite point in time.

scholarly journals Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

2019 ◽  
Vol 9 (1) ◽  
pp. 613-632 ◽  
Author(s):  
Wei Lian ◽  
Runzhang Xu
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Local Existence ◽  
High Energy ◽  
Initial Energy ◽  
Infinite Time ◽  
Logarithmic Term ◽  
Strong Damping

Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

Well-posedness and exponential stability of a flexible structure with second sound and time delay

2018 ◽  
Vol 98 (16) ◽  
pp. 2903-2915 ◽  
Author(s):  
Gang Li ◽  
Yue Luan ◽  
Jiangyong Yu ◽  
Feida Jiang
Keyword(s):  
Time Delay ◽  
Exponential Stability ◽  
Flexible Structure ◽  
Well Posedness ◽  
Second Sound

scholarly journals Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

2020 ◽  
Vol 9 (1) ◽  
pp. 1569-1591
Author(s):  
Menglan Liao ◽  
Qiang Liu ◽  
Hailong Ye
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Finite Time ◽  
Evolution Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Potential Well

Abstract In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.

Infinite time blow-up for half-harmonic map flow from R into S1

2021 ◽  
Vol 143 (4) ◽  
pp. 1261-1335
Author(s):  
Yannick Sire ◽  
Juncheng Wei ◽  
Youquan Zheng
Keyword(s):  
Blow Up ◽  
Harmonic Map ◽  
Infinite Time ◽  
Harmonic Map Flow
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