On Fisher's equation with the fractionalp‐Laplacian

Abstract

In the present note, we show that if the initial date is between zero and one, then the global solution of a Dirichlet Fisher‐KPP equation with the fractional ‐Laplacian is also between zero and one. As a consequence, a large‐time behavior of the global solution is discussed. In addition, some blow‐up results including a general Lipschitz nonlinearity are given. The proofs mainly rely on special symmetry and invariance properties of the fractional ‐Laplacian.