Existence and Smoothing for a Nondivergence-Form Degenerate Diffusion from Plasma-Wave Theory

Abstract

We prove existence and positive-time smoothing of weak solutions to the degenerate parabolic Cauchy--Dirichlet problem ${\partial }_{t}u={\rho }_{\lambda }(x)u{\partial }_x^{2}u+{\rho }_{\lambda }(x)g(x)u$ on the half-line, where ${\rho }_{\lambda }$ vanishes at the boundary and grows at infinity. This scalar problem arises from a formal reduction of the one-dimensional quasilinear system for plasma waves, which couples an electron distribution to a wave-energy spectrum: we develop a solution theory for the scalar problem that lifts to this system. For the scalar problem, we construct nonnegative weak solutions under weighted integrability assumptions on the initial data and boundedness of the reaction coefficient ${\rho }_{\lambda }g$, admitting unbounded and discontinuous data pertinent to the physical model. We establish a one-sided bound on ${\partial }_{t}u$ which yields positive-time smoothing: solutions become locally Hölder continuous in both space and time and locally Lipschitz continuous in space. Explicit examples show that this spatial regularity is sharp. Under natural assumptions on the initial data, our scalar solutions lift to weak solutions of the particle--wave system that preserve nonnegativity, conserve particle mass, smooth initial particle measures to locally bounded densities at positive times, and exhibit quantitative decay of the wave spectrum at large wavenumber.

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