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 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 Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

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>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.

scholarly journals APPROXIMATE SOLUTIONS OF SOME NONLINEAR PROBLEMS FOR THE MONGE-AMPERE EQUATIO

2020 ◽  
pp. 97-105
Author(s):  
N.B. Iskakova ◽  
◽  
А.S. Rysbek ◽  
N.S. Serik ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Initial Conditions ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Mathcad Software ◽  
The Right ◽  
Monge Ampere

Due to numerous applications in various fields of science, including gas dynamics, meteorology, differential geometry, and others, the Monge – ampere equation is one of the most intensively studied equations of nonlinear mathematical physics.In this report, we study a nonlinear boundary value problem for the inhomogeneous Monge-ampere equation, the right part of which contains power nonlinearities in derivatives and arbitrary nonlinearity from the desired function.Based on linearization, the studied boundary value problems are reduced to a system of ordinary first-order differential equations with initial conditions that depend on the parameter.Methods for constructing exact and approximate solutions of some boundary value problems for the Monge-ampere equation are proposed.Using the Mathcad software package, numerical implementation of methods for constructing approximate solutions of the obtained systems of ordinary differential equations with a parameter is performed.Three-dimensional graphs of exact and approximate solutions of the problems under consideration in the Grafikus service are constructed.

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