Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition

2019 ◽  
Vol 159 (3-4) ◽  
pp. 475-509
Author(s):  
Takayoshi Ogawa ◽  
Hiroshi Wakui
Keyword(s):  
Diffusion Equation ◽  
Critical Exponent ◽  
Finite Time ◽  
Blow Up ◽  
Uniform Bound ◽  
Drift Diffusion ◽  
Weight Condition

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.

Critical exponent in a Stefan problem with kinetic condition

1997 ◽  
Vol 8 (5) ◽  
pp. 525-532 ◽  
Author(s):  
ZHICHENG GUAN ◽  
XU-JIA WANG
Keyword(s):  
Free Boundary ◽  
Critical Exponent ◽  
Global Solution ◽  
Stefan Problem ◽  
Finite Time ◽  
Blow Up ◽  
One Dimensional ◽  
Kinetic Condition ◽  
Dirac Function ◽  
The One

In this paper we deal with the one-dimensional Stefan problemut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].

scholarly journals The critical exponent for fast diffusion equation with nonlocal source

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Chunxiao Yang ◽  
Linghua Kong ◽  
Yingxue Wu ◽  
Qing Tian
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Critical Exponent ◽  
Finite Time ◽  
Global Solutions ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Nonlocal Source ◽  
Nonlinear Anal ◽  
The Cauchy Problem

Abstract This paper considers the Cauchy problem for fast diffusion equation with nonlocal source $u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}$ u t = Δ u m + ( ∫ R n u q ( x , t ) d x ) p − 1 q u r + 1 , which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent $p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}$ p c = m + 2 q − n ( 1 − m ) − n q r n ( q − 1 ) , namely, any solution of the problem blows up in finite time whenever $1< p\le p_{c}$ 1 < p ≤ p c , and there are both global and non-global solutions if $p>p_{c}$ p > p c .

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.

scholarly journals Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Razvan Gabriel Iagar ◽  
Ana Isabel Muñoz ◽  
Ariel Sánchez
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Space Dimension ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlinear Reaction ◽  
Different Types ◽  
Self Similar

<p style='text-indent:20px;'>We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in any space dimension <inline-formula><tex-math id="M1">\begin{document}$ x\in \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula> and with exponents <inline-formula><tex-math id="M3">\begin{document}$ m&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p\in(0, 1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \sigma&gt;2(1-p)/(m-1) $\end{document}</tex-math></inline-formula>. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are <i>compactly supported</i> and might present two different types of interface behavior and three different possible <i>good behaviors</i> near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>. This paper generalizes in dimension <inline-formula><tex-math id="M7">\begin{document}$ N&gt;1 $\end{document}</tex-math></inline-formula> previous results by the authors in dimension <inline-formula><tex-math id="M8">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula> and also includes some finer classification of the profiles for <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> large that is new even in dimension <inline-formula><tex-math id="M10">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>

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