- Tytuł:
- Existence and smoothing effects of the initial-boundary value problem for ∂u/∂t−Δσ(u) = 0 in time-dependent domains
- Autorzy:
- Nakao, Mitsuhiro
- Powiązania:
- https://bibliotekanauki.pl/articles/29519532.pdf
- Data publikacji:
- 2023
- Wydawca:
- Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
- Tematy:
-
quasilinear parabolic equation
time-dependent domain
smoothing effect - Opis:
- We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form $ u_t − Δσ(u) = 0 $ in $ Q(0, T) $ with the initial and boundary conditions $ u(0) = u0 $ and $ u(t)|_{∂Ω(t)} = 0 $, where Ω(t) is a bounded domain in $ R^N $ for each t ≥ 0 and $ Q(0,T) = \bigcup_{0<t<T} \Omega(t) \times {t}, T>0 $. Our class of $ σ(u) $ includes $ σ(u) = |u|^m u $, $ σ(u) = u log(1 + |u|^m), 0 ≤ m ≤ 2, $ and $ σ(u) |u|^m u// \sqrt{1+|u|^2} $, $ 1 ≤ m ≤ 2 $ etc. We derive precise estimates for $ ||u(t)||_{Ω(t),∞} $ and $ ||∇σ(u(t))||_{Ω(t),2}^2, t > 0 $, depending on $ ||u_0||_{Ω(0),r} $ and the movement of $ ∂Ω(t) $.
- Źródło:
-
Opuscula Mathematica; 2023, 43, 5; 703-734
1232-9274
2300-6919 - Pojawia się w:
- Opuscula Mathematica
- Dostawca treści:
- Biblioteka Nauki
Artykuł