scholarly journals Prescribing Morse Scalar Curvatures: Blow-Up Analysis

2020 ◽  
Author(s):  
Andrea Malchiodi ◽  
Martin Mayer
Keyword(s):  
Critical Points ◽  
Blow Up ◽  
Finite Energy ◽  
Existence Results ◽  
Critical Regime ◽  
Geometric Problem ◽  
Critical Points At Infinity ◽  
Blow Up Analysis ◽  
Blowing Up ◽  
Subcritical Solutions

Abstract We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais–Smale sequences, we determine precise blow-up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers, we aim to establish the sharpness of this result, proving a converse existence statement, together with a one-to-one correspondence of blowing-up subcritical solutions and critical points at infinity. This analysis will be then applied to deduce new existence results for the geometric problem.

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.

Theory of “Critical Points at Infinity” and a Resonant Singular Liouville-Type Equation

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Mohameden Ahmedou ◽  
Mohamed Ben Ayed
Keyword(s):  
Variational Problem ◽  
Critical Point ◽  
Critical Points ◽  
Critical Point Theory ◽  
Type Equation ◽  
Point Theory ◽  
Existence Results ◽  
Liouville Type ◽  
Resonant Case ◽  
Critical Points At Infinity

AbstractWe consider the following Liouville-type equation on domains ofwhereUsing some dynamical and topological tools from the “critical point theory at infinity” of Bahri, we study the critical points at infinity of the related variational problem. Then we derive from our analysis some existence results in the so-called resonant case, that is, when the parameter ϱ is of the form

scholarly journals Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

2002 ◽  
Vol 2 (2) ◽  
Author(s):  
Mohamed Ben Ayed ◽  
Khalil El Mehdi ◽  
Mohameden Ould Ahmedou
Keyword(s):  
Boundary Conditions ◽  
Variational Problem ◽  
Scalar Curvature ◽  
Critical Points ◽  
Existence Results ◽  
Topological Methods ◽  
Critical Points At Infinity

AbstractThis paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

On the Boundary Behavior for the Blow-up Solutions of the sinh-Gordon Equation and Rank N Toda Systems in Bounded Domains

2018 ◽  
Vol 2020 (23) ◽  
pp. 9386-9419 ◽  
Author(s):  
Weiwei Ao ◽  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Gordon Equation ◽  
Blow Up ◽  
Boundary Behavior ◽  
Bounded Domains ◽  
Residual Mass ◽  
Blow Up Analysis ◽  
Blowing Up ◽  
General Rank ◽  
General Existence ◽  
Toda Systems

Abstract In this paper we are concerned with the blow-up analysis of two classes of problems in bounded domains arising in mathematical physics: sinh-Gordon equation and some general rank $n$ Toda systems. The presence of a residual mass in the blowing up limit makes the analysis quite delicate; nevertheless, by exploiting suitable Pohozaev identities and a detailed blow-up analysis we exclude blowup at the boundary. This is the 1st result in this direction in the presence of a residual mass. As a byproduct we obtain general existence results in bounded domains.

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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