scholarly journals Global solution to the Cauchy problem of nonlinear thermodiffusion in a solid body

2010 ◽  
Vol 37 (4) ◽  
pp. 437-458 ◽  
Author(s):  
Arkadiusz Szymaniec
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Solid Body ◽  
The Cauchy Problem

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

scholarly journals Global Analysis for Rough Solutions to the Davey-Stewartson System

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Equation ◽  
Global Solution ◽  
Global Analysis ◽  
Space Time ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
New Space ◽  
Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

scholarly journals Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Hyungjin Huh ◽  
Bora Moon
Keyword(s):  
Cauchy Problem ◽  
Total Energy ◽  
Global Solution ◽  
Gauge Condition ◽  
Global Energy ◽  
Nirenberg Inequality ◽  
The Cauchy Problem ◽  
Neutral Field ◽  
Chern Simons ◽  
Uniqueness Of A Solution

We study the Cauchy problem of the Chern-Simons-Schrödinger equations with a neutral field, under the Coulomb gauge condition, in energy space H1(R2). We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.

scholarly journals Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3627-3639 ◽  
Author(s):  
Ruizhao Zia
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Global Solution ◽  
Besov Spaces ◽  
Coupling Constant ◽  
Unique Global Solution ◽  
Small Initial Data ◽  
Nonlinear Anal ◽  
The Cauchy Problem

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant ? ?(0,1). It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in ?Hs ??Bd/2 2,1 with - d/2 < s < d2-1. In particular, if d ? 3, and taking s=0, then ?H0 ? ?Bd/2 2,1 = B d/2 2,1. Since Bt2,? ? Bd/2 2,1 if t > d/2, this result extends the work by Chen and Miao [Nonlinear Anal.,68(2008), 1928-1939].

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