scholarly journals Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak damping

2021 ◽  
Author(s):  
Menglan Liao ◽  
Zhong Tan
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Initial Data ◽  
Dirichlet Boundary Condition ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Kirchhoff Type ◽  
Weak Damping

The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator $\mathcal{L}_K$: \[u_{tt}+[u]_s^{2(\theta-1)}\mathcal{L}_Ku+a|u_t|^{m-1}u_t=b|u|^{p-1}u\] with homogeneous Dirichlet boundary condition and initial data, where $[u]^2_s$ is the Gagliardo seminorm, $a\geq 0,~b>0,~0

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals A Lotka-Volterra Competition Model with Cross-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenyan Chen ◽  
Ya Chen
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Cross Diffusion ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

scholarly journals The decay of the local energy for wave equations with discontinuous coefficients

1977 ◽  
Vol 68 ◽  
pp. 35-58 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Discontinuous Coefficients ◽  
The Other ◽  
Neumann Condition ◽  
Local Energy ◽  
Exterior Domains

The exponential decay of the local energy for wave equations in exterior domains of the odd dimensional space has been proved in [1] ~ [6] etc. under the Dirichlet boundary condition and in [5], [7] under the Neumann condition and the other conditions. In this paper, we shall consider this problem for the following equation:

scholarly journals Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Gang Li ◽  
Linghui Hong ◽  
Wenjun Liu
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
High Energy ◽  
Dirichlet Boundary Conditions ◽  
Kirchhoff Type ◽  
Nonexistence Result ◽  
Global Nonexistence ◽  
Viscoelastic Wave ◽  
Nonexistence Of Solutions

We consider viscoelastic wave equations of the Kirchhoff typeutt-M(∥∇u∥22)Δu+∫0tg(t-s)Δu(s)ds+ut=|u|p-1uwith Dirichlet boundary conditions, where∥⋅∥pdenotes the norm in the Lebesgue spaceLp. Under some suitable assumptions ongand the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense thatlim⁡t→T*-(∥u(t)∥22+∫0t∥u(s)∥22ds)=∞for some0<T*<+∞.

scholarly journals Extinction and Nonextinction for the Fast Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chunlai Mu ◽  
Li Yan ◽  
Yi-bin Xiao
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Dirichlet Boundary ◽  
Fast Diffusion ◽  
First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Fast Diffusion Equation ◽  
The First Eigenvalue

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain ofRNwithN>2. For0<m<1, under appropriate hypotheses, we show thatm=pis the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of-Δwith homogeneous Dirichlet boundary.

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