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 Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pierre Roux ◽  
Delphine Salort
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Mean Field ◽  
Classical Solutions ◽  
Field Limit ◽  
Mean Field Limit ◽  
Diffusive Approximation ◽  
The Mean ◽  
Weak Connectivity ◽  
Time Existence

<p style='text-indent:20px;'>The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.</p>

scholarly journals A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

2018 ◽  
Vol 149 (2) ◽  
pp. 325-352 ◽  
Author(s):  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Field Equation ◽  
Elliptic Problem ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Probability Measures ◽  
Trudinger Inequality ◽  
Mean Field Equation

AbstractWe are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

scholarly journals Boundary value problems for local and nonlocal Liouville type equations with several exponential type nonlinearities. Radial and nonradial solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Angela Slavova ◽  
Petar Popivanov
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Mean Field ◽  
Radial Solution ◽  
Liouville Type ◽  
The Cauchy Problem ◽  
Logarithmic Singularities ◽  
The Mean ◽  
The Right ◽  
Mean Field Equation

AbstractThis paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc $B_{1} \subset {\mathbf{R}}^{2} $ B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at $\partial B_{1} $ ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for $r = 0 $ r = 0 , $r = \infty $ r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals On the existence and blow-up of solutions for a mean field equation with variable intensities

2016 ◽  
Vol 27 (4) ◽  
pp. 413-429 ◽  
Author(s):  
Tonia Ricciardi ◽  
Ryo Takahashi ◽  
Gabriella Zecca ◽  
Xiao Zhang
Keyword(s):  
Field Equation ◽  
Blow Up ◽  
Mean Field ◽  
Mean Field Equation

scholarly journals Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Jincheng Shi ◽  
Shengzhong Xiao
Keyword(s):  
Global Existence ◽  
Life Span ◽  
General Model ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Conditions ◽  
Short Wave ◽  
Classical Solutions ◽  
Global Classical Solutions

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau&gt;0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

scholarly journals An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

2013 ◽  
Vol 143 (5) ◽  
pp. 1021-1045 ◽  
Author(s):  
Aleks Jevnikar
Keyword(s):  
Field Equation ◽  
Existence Result ◽  
Mean Field ◽  
Type Inequality ◽  
Riemannian Surface ◽  
Supercritical Regime ◽  
Supercritical Parameters ◽  
The Mean ◽  
First Time ◽  
Mean Field Equation

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean-field equation of the equilibrium turbulence with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing so, we present a new Moser–Trudinger-type inequality under suitable conditions on the centre of mass and the scale of concentration of both eu and e−u, where u is the unknown function in the equation.

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