Life-span of subcritical semilinear wave equation (Q2774031)

This paper deals with a new weighted Strichartz estimate for the classical wave equation in the case of space dimension \(n\geq 4\). The authors apply the estimate mentioned above to the study of the life-span of the solution to the Cauchy problem for a semilinear wave equation with a right-hand side \(|u|^p\), \(p>1\), the exponent \(p\) being subcritical, i.e. \(-{n-1 \over 2}p^2+ {n+1\over 2}p+1>0\), \(n\geq 4\). The Cauchy data are \(\varepsilon>0\) small, i.e. are of the type \((\varepsilon f_0,\varepsilon f_1)\), \(0<\varepsilon \ll 1\), and belong to suitable Sobolev spaces. The authors estimate from below the life span \(T(\varepsilon)\) of the weak solution of the corresponding initial value problem. The estimate from above of \(T(\varepsilon)\) can be obtained by using some blow up results of Sideris. The cases \(n=2,3\) have been studied by John, Glassey, Schaeffer, Sideris, Yi Zhou.