On the equation □φ=|delφ|2 in four space dimensions

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions squarephi = F(partial derivativephi), phi(0, x) = f(x), partial derivative(t)phi(0, x) = g(x), where square = partial derivative(t)(2) - Delta is the wave operator, F is quadratic in partial derivativephi with partial derivative = (partial derivative(t), partial derivative(x1), ..., partial derivativex(n)). The minimal value of s is determined such that the above Cauchy problem is locally well-posed in H-s. It turns out that for the general equation s must satisfy s > max(n/2, n+5/4). This is due to Ponce and Sideris (when n = 3) and Tataru (when n greater than or equal to 5). The purpose of this paper is to supplement with a proof in the case n = 2,4.