Very slow grow-up of solutions of a semi-linear parabolic equation

2011 ◽  
Vol 54 (2) ◽  
pp. 381-400 ◽  
Author(s):  
Marek Fila ◽  
John R. King ◽  
Michael Winkler ◽  
Eiji Yanagida
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Large Time ◽  
Global Solutions ◽  
Linear Parabolic Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Supercritical Nonlinearity ◽  
Singular Steady State

AbstractWe consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

scholarly journals Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Equilibrium Solution ◽  
Global Solutions ◽  
Initial Time ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Unique Global Solution ◽  
Initial Value ◽  
The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

scholarly journals On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Hatice Taskesen ◽  
Necat Polat ◽  
Abdulkadir Ertaş
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Type Equation ◽  
Global Solutions ◽  
Initial Energy ◽  
Potential Well ◽  
Global Weak Solutions ◽  
Power Type ◽  
Initial Value ◽  
The Cauchy Problem

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

scholarly journals GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CAUCHY PROBLEM OF A DISSIPATIVE BOUSSINESQ-TYPE EQUATION

2017 ◽  
Vol 22 (4) ◽  
pp. 441-463 ◽  
Author(s):  
Amin Esfahani ◽  
Hamideh B. Mohammadi
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Contraction Mapping ◽  
Type Equation ◽  
Global Solutions ◽  
Contraction Mapping Principle ◽  
Equation Modeling ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle.

scholarly journals Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density

2021 ◽  
Vol 2131 (3) ◽  
pp. 032043
Author(s):  
M Aripov ◽  
A S Matyakubov ◽  
J O Khasanov ◽  
M M Bobokandov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Variable Density ◽  
Divergence Form ◽  
Cross Diffusion ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

Abstract In this paper the properties of solutions of nonlinear parabolic equation not in divergence form | x | − 1 ∂ u ∂ t = u q ∂ ∂ x ( | x | n u m − 1 | ∂ u k ∂ x | p − 2 ∂ u ∂ x ) + | x | − 1 u β are studied. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic equation not in divergence form, depending on the value in the equation of the numerical parameters necessary and sufficient signs of their existence. The compactly supported solution of the Cauchy problem for a cross-diffusion parabolic equation not in divergence form with a source and a variable density is obtained.

Two-parameter asymptotics in the Cauchy problem for a quasi-linear parabolic equation

2009 ◽  
Vol 63 (1-2) ◽  
pp. 49-54
Author(s):  
Sergei V. Zakharov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Linear Parabolic Equation ◽  
The Cauchy Problem ◽  
Two Parameter

The existence and nonexistence of global solutions of the Cauchy problem for Klein-Gordon systems

2014 ◽  
Vol 90 (3) ◽  
pp. 680-682
Author(s):  
A. B. Aliev ◽  
A. A. Kazimov
Keyword(s):  
Cauchy Problem ◽  
Global Solutions ◽  
Existence And Nonexistence ◽  
The Cauchy Problem ◽  
Klein Gordon

scholarly journals Existence and nonexistence of global solutions to the Cauchy problem of thenonlinear hyperbolic equation with damping term

2018 ◽  
Vol 3 (2) ◽  
pp. 322-342 ◽  
Author(s):  
Jiali Yu ◽  
◽  
Yadong Shang ◽  
Huafei Di
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Global Solutions ◽  
Damping Term ◽  
Existence And Nonexistence ◽  
The Cauchy Problem

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

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