Blow-up for nonlinear wave equations with slowly decaying data
We consider classical solutions to the initial value problem for the ``accretive'' equation \(\square u = | u_ t |^ p\) in \(\mathbb{R}^ n \times [0, \infty)\) with ``small'' data. If the data have compact support, it is partially known that there is a number \(p_ 0(n)\) such that solutions blow-up in finite time for \(1< p \leq p_ 0 (n)\) and a global solution exists for \(p > p_ 0(n)\). In this paper, we shall show for \(n = 4,5\) that, if the support of data is noncompact, there are blowing-up solutions even for \(p > p_ 0 (n)\) because of the ``bad'' spatial decay of the initial data.
scientific article; zbMATH DE number 737154
scientific article