Global Existence for Semilinear Wave Equations with Scale-Invariant Damping
The sharp threshold result that the damped wave equation d_t^2 u - Delta u + (mu/t) d_t u = |u|^p admits global small-data weak solutions for n >= 3 and mu in (0,1) or (1,2) whenever p lies between the shifted Strauss exponent pcrit(n,mu), the positive root of (n+mu-1)p^2 - (n+mu+1)p - 2 = 0, and the conformal exponent (n+mu+3)/(n+mu-1). A student learns how scale-invariant damping shifts the effective dimension from n to n+mu, and how a change of variables converts the damped wave equation into a semilinear generalized Tricomi equation whose weighted Strichartz estimates drive the existence proof.
Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant
Proves a sharp global existence result for the semilinear wave equation d_t^2 u - Delta u + (mu/t) d_t u = |u|^p with time-dependent scale-invariant damping — scale-invariant because the mu/t coeffic…