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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.13922 |
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Table of Contents:
- We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem $\partial_{t}u - ρ_λ(x)uΔu = ρ_λ(x)g_{0}(x)u$ on the half line $(0,\infty)$. We find weak solutions from $L^p\ (p < \infty)$ initial data, and by means of a Benilan-Crandall inequality, show solutions are jointly Holder continuous, and locally, spatially Lipschitz on the parabolic interior. We identify special solutions which saturate these bounds. The Benilan-Crandall inequality, derived from time-scaling arguments, is of independent interest for exposing a regularizing effect of the parabolic u$Δ$u operator. Recently considered in [11], this problem originates in the theory of nonlinear instability damping via wave-particle interactions in plasma physics (see [8, 22]).