Failure of property (P) for nonlinearities in two dimensions

Determine whether there exists a superlinear nonlinearity for which property (P)—eventual nondecrease of the L-infinity norm whenever it is sufficiently large, with the threshold depending only on the initial-data bound, the nonlinearity, and the domain—fails in two spatial dimensions.

Background

Property (P) asserts that, for a prescribed bound on the initial L-infinity norm, sufficiently large values of the solution norm force the norm to remain nondecreasing thereafter. The paper proves failure of this property in several supercritical settings, while establishing it for subcritical power nonlinearities. The authors note that the analogous existence question for superlinear nonlinearities in dimension two remains unresolved.