Finite time blow up for a solution to system of the drift–diffusion equations in higher dimensions

2016 ◽  
Vol 284 (1-2) ◽  
pp. 231-253 ◽  
Author(s):  
Masaki Kurokiba ◽  
Takayoshi Ogawa
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Higher Dimensions ◽  
Diffusion Equations ◽  
Drift Diffusion

scholarly journals Blow-up for discrete reaction-diffusion equations on networks

2015 ◽  
Vol 9 (1) ◽  
pp. 103-119 ◽  
Author(s):  
Soon-Yeong Chung ◽  
Jae-Hwang Lee
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Blow Up Rate

In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.

Non-uniform bound and finite time blow up for solutions to a drift–diffusion equation in higher dimensions

2016 ◽  
Vol 14 (01) ◽  
pp. 145-183 ◽  
Author(s):  
Takayoshi Ogawa ◽  
Hiroshi Wakui
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Blow Up ◽  
Elliptic Type ◽  
Sharp Constant ◽  
Uniform Bound ◽  
Critical Space ◽  
Drift Diffusion ◽  
Entropy Functional ◽  
Blowing Up

We show the non-uniform bound for a solution to the Cauchy problem of a drift–diffusion equation of a parabolic–elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at [Formula: see text] in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.

scholarly journals Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

2020 ◽  
Vol 482 (1) ◽  
pp. 123503 ◽  
Author(s):  
Benito Hernández-Bermejo ◽  
Razvan Gabriel Iagar ◽  
Pilar R. Gordoa ◽  
Andrew Pickering ◽  
Ariel Sánchez
Keyword(s):  
Finite Time ◽  
Nonlinear Diffusion ◽  
Blow Up ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations

LOCAL WELL POSEDNESS OF CAUCHY PROBLEM FOR VISCOUS DIFFUSION EQUATIONS

2009 ◽  
Vol 20 (04) ◽  
pp. 509-519
Author(s):  
YACHENG LIU ◽  
RUNZHANG XU
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Finite Time ◽  
Blow Up ◽  
Diffusion Equations ◽  
Well Posedness ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem of multi-dimensional viscous diffusion equations. By using an equivalent integral equations, we get the existence of local Wk,p solutions. And we prove the finite time blow up of solutions under appropriate conditions.

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