Kaydedildi:
| Asıl Yazarlar: | , , |
|---|---|
| Materyal Türü: | Preprint |
| Baskı/Yayın Bilgisi: |
2025
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| Konular: | |
| Online Erişim: | https://arxiv.org/abs/2508.06384 |
| Etiketler: |
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İçindekiler:
- Given a parabolic cylinder $Q =(0,T)\timesΩ$, where $Ω\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-Δ_p u = μ\quad \text{in $Q$} \] with Dirichlet boundary conditions, where $μ$ is a finite Radon measure in $Q$. We first prove a priori estimates on the $p$-parabolic capacity of level sets of $u$. We then show that diffuse measures (i.e.\@ measures which do not charge sets of zero parabolic $p$-capacity) can be strongly approximated by the measures $μ_k = (T_k(u))_t-Δ_p(T_k(u))$, and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of $$ u_t-Δ_{p} u + h(u)=μ\quad \text{in $Q$,} $$ for any function $h$ such that $h(s)s\geq 0$ and for any diffuse measure $μ$; when $h$ is nondecreasing we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.