Delayed singularity formation for solutions of nonlinear partial differential equations in higher dimensions.
Strict solutions u of genuinely nonlinear homogeneous hyperbolic equations in two independent variables with initial data f(x) of compact support become singular after a time interval of order parallelf parallel(-1). In higher dimensions solutions initially of compact support are likely to have life expectancies of orders parallelf parallel(-2+epsilon) at least. This is proved for the special case of solutions u(x(1),..., x(n), t) of a second order equation u(tt) = Sigma(i,j)a(ij)u(xixj), where n >/= 3 and where the coefficients a(ij) are C(infinity)-functions in the first derivatives of u, forming a symmetric positive definite matrix.
John
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Courant Institute of Mathematical Sciences, New York University, New York, N.Y. 10012.