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The local strong solutions and global weak solutions for a nonlinear equation. (English) Zbl 1291.35315
Summary: The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space \(C([0, T); H^s(\mathbb{R})) \cap C^1([0, T); H^{s-1}(\mathbb{R}))\) provided that the initial value lies in \(H^s(\mathbb{R})\) with \(s > 3/2\). Meanwhile, we prove the existence of global weak solutions in \(L^\infty([0, \infty); L^2(\mathbb{R}))\) for the equation.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35D30 Weak solutions to PDEs
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