Opuscula Math. 43, no. 5 (2023), 703-734
https://doi.org/10.7494/OpMath.2023.43.5.703
Opuscula Mathematica
Existence and smoothing effects of the initial-boundary value problem for \(\partial u/\partial t-\Delta\sigma(u)=0\) in time-dependent domains
Abstract. 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 \[u_t-\Delta \sigma(u) =0 \quad \text{in } Q(0, T)\] with the initial and boundary conditions \[u(0)=u_0 \quad \text{and} \quad u(t)|_{\partial \Omega(t)}=0,\] where \(\Omega(t)\) is a bounded domain in \(R^N\) for each \(t \geq 0\) and \[Q(0,T)=\bigcup_{0 \lt t \lt T} \Omega(t) \times \{t\}, \quad T>0.\] Our class of \(\sigma(u)\) includes \(\sigma(u)=|u|^m u\), \(\sigma(u)=u \log (1+ |u|^m)\), \(0\leq m \leq 2\), and \(\sigma(u)=|u|^{m}u/\sqrt{1+|u|^2}\), \(1 \leq m \leq 2\), etc. We derive precise estimates for \(\|u(t)\|_{\Omega(t),\infty}\) and \(\|\nabla\sigma(u(t))\|^2_{\Omega(t),2}\), \(t\gt 0\), depending on \(\|u_0\|_{\Omega(0),r}\) and the movement of \(\partial\Omega(t)\).
Keywords: quasilinear parabolic equation, time-dependent domain, smoothing effects.
Mathematics Subject Classification: 35B40, 35K92.
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- Mitsuhiro Nakao
- Faculty of Mathematics, Kyushu University, Moto-oka 819-1602, Fukuoka, Japan
- Communicated by J.I. Díaz.
- Received: 2022-12-07.
- Revised: 2023-05-05.
- Accepted: 2023-05-09.
- Published online: 2023-06-24.
