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Low regularity semilinear wave equations. (English) Zbl 0939.35123
The author studies the Cauchy problem for the semilinear wave equation: \[ u_{tt}- \Delta u= F(u),\quad u(0, \cdot)=f \in H^\gamma ({\mathbb R}^n),\quad \partial_tu(0, \cdot)=g \in H^{\gamma -1} ({\mathbb R}^n), \] where \( n > 3\), and the nonlinear term satisfies \(F(0)=0\), \( |F(u)-F(v)|\leq |u-v|(|u|^{p-1}+|v|^{p-1})\). Under suitable conditions on \(\gamma, p, n\) the local well-posedness is proved for \( 0< \gamma < {n-3\over 2(n-1)}\).

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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