Nonlinear wave equations on the two-dimensional sphere
This paper is devoted to an initial value problem on the two-dimensional sphere \(S^2\), that is \[ \begin{cases} u_{tt}-\Delta u=G(u),\\ u(x,0)=u_0,\;u_t(x,0)= u_1,\end{cases}\tag{1} \] where \(\Delta\) is the Laplace-Beltrami operator on \(S^2\). Under some suitable assumptions on \(G(u)\) the authors prove existence and uniqueness of a local regular solution. Morevoer they show that a large class of solutions to the initial value problem quench in finite time.
scientific article; zbMATH DE number 1749659
scientific article