Displaying similar documents to “Global existence for the nonlinear equations of crystal optics”

Continuous dependence and general decay of solutions for a wave equation with a nonlinear memory term

Doan Thi Nhu Quynh, Nguyen Huu Nhan, Le Thi Phuong Ngoc, Nguyen Thanh Long (2023)

Applications of Mathematics

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We study existence, uniqueness, continuous dependence, general decay of solutions of an initial boundary value problem for a viscoelastic wave equation with strong damping and nonlinear memory term. At first, we state and prove a theorem involving local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the kernel function and the nonlinear terms. Finally, under suitable...

On the nonlinear stabilization of the wave equation

Aissa Guesmia (1998)

Annales Polonici Mathematici

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We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.

The wave map problem. Small data critical regularity

Igor Rodnianski (2005-2006)

Séminaire Bourbaki

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The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is ( 2 + 1 ) , the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory....

On bilinear estimates for wave equations

Sergiù Klainerman, Damiano Foschi (1999)

Journées équations aux dérivées partielles

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I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the L 2 theory, which is now quite well developed, I plan to discuss a more general point of view concerning the L p theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also...