Blow-up of solutions to critical semilinear wave equations with variable coefficients

Abstract

We verify the critical case p=p0(n) of Strauss' conjecture [30] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in Rn, where n≥2. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when p=p0(n). The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [43] and Zhou & Han [45]: exponential “eigenfunctions” of the Laplacian [37] are used to construct the test function ϕq for linear wave equation with variable coefficients and John's method of iterations [13] is augmented with the “slicing method” of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.