Existence of Local Solutions of Nonlinear Wave Equations in $n$-Dimensional Space

Abstract

In this paper, we investigate the local existence of solutions in $H^s$ for $n$-dimensional nonlinear wave equations with special nonlinear terms, such as $$u_{tt}-\Delta u=u^k|\nabla u|^l,\ \ x\in R^n,\ \ k\in Z^+,\ \ l\geq2.$$ where $\nabla u=(\frac{\partial u}{\partial x_1},\ \frac{\partial u}{\partial x_2},\ \cdots,\ \frac{\partial u}{\partial x_n})$. Meanwhile, we obtain that the regular index $s$ of Sobolev space $H^s$ satisfies $s>\max\{\frac{n+5}4;\ \frac n2+1-\frac1{l-1}\},\ n>3$.