  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.
