Localization Regimes for Quasilinear Parabolic p-Laplacian Equations under Unbounded Boundary Forcing

This study formulates an open problem on spatial localization for quasilinear parabolic equations of pLaplacian type on the half-space R+N, with smooth, nonnegative initial data and boundary conditions that grow unbounded over time. While localization has been widely studied under homogeneous or decaying boundary conditions, its persistence under unbounded boundary input remains unresolved. We introduce two forms of spatial localization: effective localization, where the solution remains within a finite spatial domain, and strict localization, where it vanishes beyond a fixed boundary. The problem examines how the structure of the nonlinearities A(v) and B(v), and the growth rate of the boundary function ϕ influence solution confinement or spread. The study extends classical localization theory to degenerate and singular diffusion processes with diffusion exponent p > 1, offering new insight into spatial confinement under persistent external forcing.