scholarly journals Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

2017 ◽  
Vol 20 (02) ◽  
pp. 1750026 ◽  
Author(s):  
Ali Hyder ◽  
Stefano Iula ◽  
Luca Martinazzi
Keyword(s):  
Euclidean Space ◽  
Liouville Equation ◽  
Blow Up ◽  
Positive Function ◽  
Bounded Sequence ◽  
Open Domain ◽  
Curvature Equation ◽  
Blowing Up ◽  
Sequence Of Functions

Let [Formula: see text] be an integer. For any open domain [Formula: see text], non-positive function [Formula: see text] such that [Formula: see text], and bounded sequence [Formula: see text] we prove the existence of a sequence of functions [Formula: see text] solving the Liouville equation of order [Formula: see text] [Formula: see text] and blowing up exactly on the set [Formula: see text], i.e. [Formula: see text] thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of [Formula: see text] and to the case [Formula: see text]. Several related problems remain open.

scholarly journals Blowing-up solutions for a supercritical elliptic equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yessine Dammak
Keyword(s):  
Blow Up ◽  
Positive Function ◽  
Dimensional System ◽  
Positive Real ◽  
Subcritical Case ◽  
Smooth Bounded Domain ◽  
Finite Dimensional ◽  
Blowing Up ◽  
Two Bubbles ◽  
Blowing Up Solutions

<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>

scholarly journals A quasi-linear parabolic system of chemotaxis

2006 ◽  
Vol 2006 ◽  
pp. 1-21 ◽  
Author(s):  
Takasi Senba ◽  
Takasi Suzuki
Keyword(s):  
Diffusion Coefficient ◽  
Euclidean Space ◽  
Bounded Domain ◽  
Parabolic System ◽  
Positive Constant ◽  
Diffusion Coefficients ◽  
Finite Time ◽  
Blow Up ◽  
Positive Function ◽  
Dimensional Euclidean Space

We consider a quasi-linear parabolic system with respect to unknown functionsuandvon a bounded domain ofn-dimensional Euclidean space. We assume that the diffusion coefficient ofuis a positive smooth functionA(u), and that the diffusion coefficient ofvis a positive constant. IfA(u)is a positive constant, the system is referred to as so-called Keller-Segel system. In the case where the domain is a bounded domain of two-dimensional Euclidean space, it is shown that some solutions to Keller-Segel system blow up in finite time. In three and more dimensional cases, it is shown that solutions to so-called Nagai system blow up in finite time. Nagai system is introduced by Nagai. The diffusion coefficients of Nagai system are positive constants. In this paper, we describe that solutions to the quasi-linear parabolic system exist globally in time, if the positive functionA(u)rapidly increases with respect tou.

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.

scholarly journals Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 639-675
Author(s):  
Thierry Cazenave ◽  
Yvan Martel ◽  
Lifeng Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Finite Energy ◽  
Energy Solution ◽  
Compact Set ◽  
Existence Of A Solution ◽  
Blowing Up ◽  
Finite Energy Solution

AbstractWe prove that any sufficiently differentiable space-like hypersurface of {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation {\partial_{tt}u-\Delta u=|u|^{p-1}u} on {{\mathbb{R}}\times{\mathbb{R}}^{N}}, for any {1\leq N\leq 4} and {1<p\leq\frac{N+2}{N-2}}. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at {t=0}. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with {H^{2}\times H^{1}} solutions for the transformed problem.

scholarly journals On Blowing Up the Pokies: The Pokie Lounge as a Cultural Site of Neoliberal Governmentality in Australia

2011 ◽  
Vol 17 (2) ◽  
Author(s):  
Fiona Nicoll
Keyword(s):  
Blow Up ◽  
Public Affairs ◽  
Cultural Dynamics ◽  
Neoliberal Governmentality ◽  
Television Shows ◽  
Self Help ◽  
Blowing Up ◽  
Fatal Attraction ◽  
Commercial Radio ◽  
Gaming Machines

In 1999 The Whitlams, a popular ‘indie’ band named after a former Australian prime minister whose government was controversially sacked in 1975 by the Governor-General, released a single titled ‘Blow up the Pokies’. Written about a former band member’s fatal attraction to electronic gaming machines (henceforth referred to as ‘pokies’), the song was mixed by a top LA producer, a decision that its writer and The Whitlam’s front-man, Tim Freedman, describes as calculated to ‘get it on big, bombastic commercial radio’. The investment paid off and the song not only became a big hit for the band, it developed a legacy beyond the popular music scene, with Freedman invited to write the foreword of a ‘self-help manual for giving up gambling’ as well as appearing on public affairs television shows to discuss the issue of problem gambling. The lyrics of ‘Blow up the Pokies’ frame the central themes of this article: spaces, technologies and governmentality of gambling. It then explores what cultural articulations of resistance to the pokie lounge tell us about broader social and cultural dynamics of neoliberal governmentality in Australia.

scholarly journals The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

2008 ◽  
Vol 145 (3) ◽  
pp. 643-667 ◽  
Author(s):  
DANIELA KRAUS ◽  
OLIVER ROTH
Keyword(s):  
Continuous Function ◽  
Liouville Equation ◽  
Gaussian Curvature ◽  
Boundary Point ◽  
Classical Result ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Isolated Singularities ◽  
Hölder Continuous ◽  
Curvature Equation

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

scholarly journals Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

2019 ◽  
Vol 2019 (754) ◽  
pp. 225-251 ◽  
Author(s):  
James Isenberg ◽  
Haotian Wu
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type Ii ◽  
Matched Asymptotics ◽  
Curve Shortening Flow ◽  
Initial Hypersurface ◽  
Rate Dependent ◽  
Blowing Up

Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

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