scholarly journals Dynamics of solutions to a semilinear plate equation with memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinxing Liu ◽  
Xiongrui Wang ◽  
Jun Zhou ◽  
Xu Liu
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plate Equation ◽  
Energy Estimation ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value ◽  
With Memory

<p style='text-indent:20px;'>In this paper we consider an initial-boundary value problem of a semilinear regularity-loss-type plate equation with memory in a bounded domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ n = 1,2,\cdots $\end{document}</tex-math></inline-formula>). By using the Faedo-Galërkin method and some theories of ordinary differential equations, we obtain the local existence and uniqueness of weak solutions. Then, we study the dynamics of the weak solutions, such as global existence and finite time blow-up, by energy estimation and some ordinary differential inequalities. Moreover, the upper bound of blow-up time for the blow-up solutions is also considered.</p>

scholarly journals Global Existence and Blow-Up for the Euler-Bernoulli Plate Equation with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Jianghao Hao ◽  
Jie Lan
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Stability Of Solutions ◽  
Plate Equation ◽  
Existence And Stability ◽  
Initial Boundary Value

We prove the local existence, blow-up, global existence, and stability of solutions to the initial boundary value problem for Euler-Bernoulli plate equation with variable coefficients.

INITIAL-BOUNDARY VALUE PROBLEMS AND FORMATION OF SINGULARITIES FOR ONE-DIMENSIONAL NON-RELATIVISTIC RADIATION HYDRODYNAMIC EQUATIONS

2012 ◽  
Vol 09 (04) ◽  
pp. 711-738 ◽  
Author(s):  
PENG JIANG ◽  
YAGUANG WANG
Keyword(s):  
Smooth Solution ◽  
Blow Up ◽  
Boundary Value ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boltzmann Equations ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value

In this paper, we investigate the well-posedness for the Euler–Boltzmann equations of radiation hydrodynamics in one spatial variable. We obtain the local existence and uniqueness of smooth solution to the initial-boundary value problem. Then, we show that a smooth solution will blow up in finite time regardless of the size of the initial disturbance.

scholarly journals Local and Global Existence of Solution for Love Type Waves with Past History

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1998
Author(s):  
Mohamed Biomy ◽  
Khaled Zennir ◽  
Ahmed Himadan
Keyword(s):  
Existence And Uniqueness ◽  
Past History ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linearization Method ◽  
Weak Compactness ◽  
Existence Of Solution ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value

In this paper, we consider an initial boundary value problem for nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.

scholarly journals On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation

2007 ◽  
Vol 38 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Shun-Tang Wu ◽  
Long-Yi Tsai
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Wave Equations ◽  
Banach Fixed Point Theorem ◽  
Kirchhoff Type ◽  
Initial Boundary Value

The initial boundary value problem for systems of nonlinear wave equations of Kirchhoff type with strong dissipation in a bounded domain is considered. We prove the local existence of solutions by Banach fixed point theorem and blow-up of solutions by energy method. Some estimates for the life span of solutions are given.

scholarly journals Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Global Weak Solutions ◽  
Singular Initial Value Problem ◽  
Derivative Term ◽  
Initial Boundary Value

We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term. In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval.

EXISTENCE AND BLOW-UP OF SOLUTIONS TO A PARABOLIC EQUATION WITH NONSTANDARD GROWTH CONDITIONS

2018 ◽  
Vol 99 (2) ◽  
pp. 242-249
Author(s):  
YANG LIU
Keyword(s):  
Parabolic Equation ◽  
Blow Up ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Growth Conditions ◽  
Initial Energy ◽  
Nonstandard Growth ◽  
Nonstandard Growth Conditions ◽  
Initial Boundary Value

We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.

scholarly journals STOCHASTIC WAVE EQUATIONS WITH NONLINEAR DAMPING AND SOURCE TERMS

2013 ◽  
Vol 16 (02) ◽  
pp. 1350013 ◽  
Author(s):  
HONGJUN GAO ◽  
FEI LIANG ◽  
BOLING GUO
Keyword(s):  
Wave Equations ◽  
Blow Up ◽  
Energy Inequality ◽  
Local Existence ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Damping ◽  
Local Existence And Uniqueness ◽  
Stochastic Wave Equations

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term |ut|q–2utand a source term of the type|u|p–2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for q ≥ p. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

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

Sign in / Sign up

Export Citation Format

Share Document