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

Abstract

We verify the critical case $p=p_0(n)$ of Strauss' conjecture (1981) concerning the blow-up of solutions to semilinear wave equations with variable coefficients in $\mathbf{R}^n$, where $n\geq 2$. 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 $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou(2007) and Zhou and Han (2014): exponential "eigenfunctions" of the Laplacian are used to construct the test function $\phi_q$ for linear wave equation with variable coefficients and John's method of iterations (1979) is augmented with the "slicing method" of Agemi, Kurokawa and Takamura (2000) for lower bounds in the critical case.