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In this paper we prove a local existence theorem for a Cauchy problem associated to a semi linear wave equation with an exponential nonlinearity in two dimension space. In this problem, the first Cauchy data is equal to zero, the second is in $L^{2} (R^{2})$ , radially symmetric and compactly supported. To prove this theorem, we first show a Moser-Trudinger type inequality for the linear problem and then we use a fixed point method to achieve the proof of the result.

Local existence of a solution of a semi-linear wave equation in a neighborhood of initial characteristic hypersurfaces

Aurore Cabet (2003)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Local solution of Carrier's equation in a noncylindrical domain.

Rabello, Tania Nunes, de Campos Vieira, Maria Cristina (2005)

Journal of Inequalities and Applications [electronic only]

Long-time behavior of nonlinear elastic waves

Thomas C. Sideris (1995)

Journées équations aux dérivées partielles

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Albert J. Milani, Hans Volkmer (2011)

Applications of Mathematics

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $u_{t t} + 2 u_{t} - a_{i j} (u_{t}, \nabla u) \partial_{i} \partial_{j} u = f$ corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation $- a_{i j} (0, \nabla v) \partial_{i} \partial_{j} v = h .$ We then give conditions for the convergence, as $t \to \infty$ , of the solution of the evolution equation to its stationary state.

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$L^{p}$ -estimates for hyperbolic operators applications to $□ u = u^{k}$

$L^{p}$ estimates for the wave equation and applications

La condition nulle pour les équations hyperboliques en dimension deux d’espace

Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation.

Large energy simple modes for a class of Kirchhoff equations.

L’équation de Klein Gordon à données petites

L’existence d’une solution classique d’une éqéaire dans $E_{1}$ (Communication préalable)

Local existence and estimations for a semilinear wave equation in two dimension space

Local existence of a solution of a semi-linear wave equation in a neighborhood of initial characteristic hypersurfaces

Local solution of Carrier's equation in a noncylindrical domain.

Long-time behavior of nonlinear elastic waves

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Low regularity and local well-posedness for the $1 + 3$ dimensional Dirac-Klein-Gordon system.

Low regularity solutions for Dirac-Klein-Gordon equations in one space dimension.

Low regularity solutions of the Chern-Simons-Higgs equations in the Lorentz gauge.

Low regularity well-posedness for the one-dimensional Dirac-Klein-Gordon system.

Lyapunov center theorem for some nonlinear PDE's : a simple proof

Currently displaying 1 – 17 of 17