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For all data in the weighted Sobolev space $H^{k}(w_{\\lambda,\\kappa}) \\cap L^{\\infty},$ with $k=\\max(0,3/2-\\alpha)$ and $w_{\\lambda, \\kappa}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $\\alpha \\in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{\\lambda, \\kappa})\\cap L^{\\infty}$ in the supercritical case $\\alpha \\in (0,1)$. 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