Abstract
We study the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution $\{V_{\gamma}*(w(t))^{2}\}w(t)$, where $V_{\gamma}(x)=|x|^{-\gamma}$, in $(x,t) \in \mathbf{R}^{n}\times \mathbf{R}$. We prove the existence of weak solutions for $0 \lt \gamma \lt$. We also prove that for $0\lt\gamma\lt{\rm Min}\{4, n\}$ the weak solution is unique and there exists a regular solution.
Citation
Takahiro Motai. "On the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution." Tsukuba J. Math. 12 (2) 353 - 369, December 1988. https://doi.org/10.21099/tkbjm/1496160835
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