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Periodic conservative solutions of the Camassa–Holm equation

Helge Holden, Xavier Raynaud (2008)

Annales de l’institut Fourier

We show that the periodic Camassa–Holm equation u t - u x x t + 3 u u x - 2 u x u x x - u u x x x = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u | t = 0 in H per 1 . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with μ ac = ( u 2 + u x 2 ) d x . The total energy is preserved by the solution.

Periodic parabolic problems with nonlinearities indefinite in sign.

Tomás Godoy, Uriel Kaufmann (2007)

Publicacions Matemàtiques

Let Ω ⊂ RN be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) up - b (x, t) uq in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can...

Periodic problems and problems with discontinuities for nonlinear parabolic equations

Tiziana Cardinali, Nikolaos S. Papageorgiou (2000)

Czechoslovak Mathematical Journal

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that...

Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems

Mihai Bostan, Eric Sonnendrücker (2006)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the existence of spatial periodic solutions for nonlinear elliptic equations - Δ u + g ( x , u ( x ) ) = 0 , x ∈ ℝ N where g is a continuous function, nondecreasing w.r.t. u . We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations....

Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems

Mihai Bostan, Eric Sonnendrücker (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the existence of spatial periodic solutions for nonlinear elliptic equations - Δ u + g ( x , u ( x ) ) = 0 , x ∈ ℝ N where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations. ...

Periodic solutions of a nonlinear evolution problem

Nelson Nery Oliveira Castro, Nirzi G. de Andrade (2002)

Applications of Mathematics

In this paper we prove existence of periodic solutions to a nonlinear evolution system of second order partial differential equations involving the pseudo-Laplacian operator. To show the existence of periodic solutions we use Faedo-Galerkin method with a Schauder fixed point argument.

Periodic solutions of a three-species periodic reaction-diffusion system

Tiantian Qiao, Jiebao Sun, Boying Wu (2011)

Annales Polonici Mathematici

We study a periodic reaction-diffusion system of a competitive model with Dirichlet boundary conditions. By the method of upper and lower solutions and an argument similar to that of Ahmad and Lazer, we establish the existence of periodic solutions and also investigate the stability and global attractivity of positive periodic solutions under certain conditions.

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