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.

EXPLOSIVE INSTABILITY DUE TO 3-WAVE OR 4-WAVE MIXING, WITH OR WITHOUT DISSIPATION

2008 ◽  
Vol 06 (04) ◽  
pp. 413-428 ◽  
Author(s):  
HARVEY SEGUR
Keyword(s):  
Initial Data ◽  
Nonlinear Waves ◽  
Finite Time ◽  
Blow Up ◽  
Wave Mixing ◽  
Explosive Instability ◽  
Effective Threshold ◽  
Blowing Up ◽  
Gain Energy

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

scholarly journals Blow-Up Criteria for the Modified Novikov Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Caochuan Ma ◽  
Wujun Lv
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Finite Time ◽  
Blow Up ◽  
Solution Blowing ◽  
Blowing Up ◽  
The Cauchy Problem ◽  
Novikov Equation

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

scholarly journals Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

2020 ◽  
Vol 20 (4) ◽  
pp. 867-894 ◽  
Author(s):  
Razvan Gabriel Iagar ◽  
Ariel Sánchez
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Blow Up ◽  
Reaction Diffusion ◽  
The Self ◽  
Reaction Diffusion Equation ◽  
Critical Range ◽  
Self Similar Solution ◽  
Self Similar ◽  
Interface Equations

AbstractWe study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight:\partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p},posed for {x\in\mathbb{R}}, {t\geq 0}, where {m>1}, {0<p<1} and {\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for {m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when {p<1}. We moreover prove that, if the condition {m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if {m+p<2}, while the critical range {m+p=2} with {\sigma>2} is postponed to a different work due to significant technical differences.

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