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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 $L^2\left({\mathbb{R}}^2\right)$, 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.