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Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008),449-475.), firstly, we construct the local strong solution to the cubicnonlinear wave equation with random data for a large set of initial data in$H^{s}(M)$ with $s\geq \frac{5}{14}$, where M is a three dimensional compactmanifold with boundary, moreover, our result improves the result of Theorem 2in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strongsolution to the quintic nonlinear wave equation with random data for a largeset of initial data in $H^{s}(M)$ with $s\geq\frac{1}{6}$, where M is a twodimensional compact boundaryless manifold; finally, we construct the localstrong solution to the quintic nonlinear wave equation with random data for alarge set of initial data in $H^{s}(M)$ with $s\geq \frac{23}{90}$, where M isa two dimensional compact manifold with boundary.

Random data Cauchy problem for the wave equation on compact manifold