On similarity solutions and blow-up spectra for a semilinear wave equation

Abstract

We construct countable spectra of different asymptotic patterns of self-similar and approximate self-similar types for global and blow-up solutions for the semilinear wave equation \[ u t t = Δ u + | u | p − 1 u , x ∈ R N , t > 0 , {u_{tt}} = \Delta u + {\left | u \right |^{p - 1}}u, \qquad x \in {R^N}, t > 0, \] in different ranges of exponent p p and dimension N N .