On well-posedness of the semilinear heat equation on the sphere (Q692806)

The author studies the Dirichlet problem of the following semilinear parabolic equation on a bounded domain (half upper sphere) on round sphere: \[ \frac{\partial u}{\partial t}=\Delta u+u^{\nu}. \] The main results are existence and uniqueness of classical solutions and non-uniqueness of singular solutions. More specifically, the authors shows, among others, that 1. If the initial data satisfies a \(L^p\) bound for appropriate \(p\), then the Dirichlet problem has unique solution \(u(t)\) for some time. 2. By comparing with the semilinear elliptic equation, non-uniqueness is shown if \(\nu=n/(n-2)\) for \(n\geq 3\). The proof relies on detailed analysis on semilinear elliptic equation and some general abstract semigroup theory for parabolic equations.