Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity

Abstract

We prove the existence and uniqueness of global solutions for a Cauchy problem associated to a semilinear Klein-Gordon equation in two space dimensions. Our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method.

Double logarithmic inequality with a sharp constant

“…Of course, we have similar inequalities for the Log Log inequality (1.2) in R 2 with the sharp constant is crucial for local wellposedness results (see [7] for further discussion). In particular from Corollary 5.2 we can derive a Moser-Trudinger type inequality for the solution of the linear Klein-Gordon.…”

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

“…For the study of the Cauchy problem in the critical case, we refer to the papers [12,21]. Under our assumptions, it is not hard to see that the proofs of [7, Theorem 6.2.2 and Proposition 6.2.3] can be adapted to give Proposition 2.…”

Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions

“…there exists a unique, smooth solution u = u(t, |x|) to the Cauchy problem (1), (2), defined for all time.…”

“…The work [2] of Ibrahim, Majdoub, and Masmoudi thus shows that the Cauchy problem for Eq. (1) is well-posed in the subcritical and critical regimes, in agreement with the known results for nonlinear wave equations…”