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.

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>

Existence and Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

2013 ◽  
Vol 785-786 ◽  
pp. 1454-1458
Author(s):  
Yan Ping Ran ◽  
Cong Ming Peng
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Equation ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

This article considers the following degenerate semilinear parabolic initial-boundary value problem,where be constants. We obtained the conditions of global existence and blow-up.

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.

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 Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity

2021 ◽  
Author(s):  
Fugeng Zeng ◽  
Qigang Deng ◽  
Dongxiu Wang
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Potential Well ◽  
Laplacian Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.

A Class of Fourth Order Damped Wave Equations with Arbitrary Positive Initial Energy

2018 ◽  
Vol 62 (1) ◽  
pp. 165-178
Author(s):  
Yang Liu ◽  
Jia Mu ◽  
Yujuan Jiao
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

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.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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.

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