On the Blow-Up of Solutions to Liouville-Type Equations

2016 ◽  
Vol 16 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Tonia Ricciardi ◽  
Gabriella Zecca
Keyword(s):  
Blow Up ◽  
Complex Structures ◽  
Liouville Type ◽  
Quick Proof

AbstractWe estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.

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.

Uniformly Elliptic Liouville Type Equations Part II: Pointwise Estimates and Location of Blow up Points

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
Daniele Bartolucci ◽  
Luigi Orsina
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Pointwise Estimate ◽  
Divergence Form ◽  
Type Problem ◽  
Pointwise Estimates ◽  
Liouville Type ◽  
Priori Estimates

AbstractWe refine the analysis, initiated in [3], [4] of the blow up phenomenon for the following two dimensional uniformly elliptic Liouville type problem in divergence form:We provide a partial generalization of a result of Y.Y. Li [18] to the case A ≠ I. To this end, in the same spirit of [2], we obtain a sharp pointwise estimate for simple blow up sequences. Moreover, we prove that if {p(∆detA)(pj) = 0, ∀ j = 1, ...,N.This characterization of the blow up set yields an improvement of the a priori estimates already established in [3].

scholarly journals Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

2020 ◽  
Author(s):  
Weiwei Ao ◽  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Mean Field ◽  
Liouville Type ◽  
Subcritical Case ◽  
Toda Systems ◽  
The Mean ◽  
Trudinger Inequality ◽  
Mean Field Equation

Abstract We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

scholarly journals Blow-up analysis of a nonlocal Liouville-type equation

2015 ◽  
Vol 8 (7) ◽  
pp. 1757-1805 ◽  
Author(s):  
Francesca Da Lio ◽  
Luca Martinazzi ◽  
Tristan Rivière
Keyword(s):  
Blow Up ◽  
Type Equation ◽  
Liouville Type ◽  
Blow Up Analysis
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