BREZIS–MERLE INEQUALITIES AND APPLICATION TO THE GLOBAL EXISTENCE OF THE CAUCHY PROBLEM OF THE KELLER–SEGEL SYSTEM

2011 ◽  
Vol 13 (05) ◽  
pp. 795-812 ◽  
Author(s):  
TOSHITAKA NAGAI ◽  
TAKAYOSHI OGAWA
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Parabolic Equations ◽  
Parabolic Systems ◽  
Unique Global Solution ◽  
Drift Diffusion ◽  
Nonlinear Parabolic ◽  
Elliptic And Parabolic Equations ◽  
The Cauchy Problem ◽  
Two Space Dimensions

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in [Formula: see text]. The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

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 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].

GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS AND DIFFUSION PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150001 ◽  
Author(s):  
YANA BELOPOLSKAYA ◽  
WOJBOR A. WOYCZYNSKI
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Diffusion Processes ◽  
Stochastic Approach ◽  
Generalized Solution ◽  
Nonlinear Parabolic Equations ◽  
Kolmogorov Equation ◽  
Generalized Solutions ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, we use results from Kunita's theory of stochastic flows.

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