TY  - JOUR
ID  - Nakao2023
LB  - Nakao2023
AU  - Nakao, Mitsuhiro
TI  - Existence and smoothing effects of the initial-boundary value problem for \(\partial u/\partial t-\Delta\sigma(u)=0\) in time-dependent domains
ST  - Existence and smoothing effects of the initial-boundary value problem for \(\partial u/\partial t-\Delta\sigma(u)=0\) in time-dependent domains
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 5
SP  - 703
EP  - 734
DO  - https://doi.org/10.7494/OpMath.2023.43.5.703
UR  - https://doi.org/10.7494/OpMath.2023.43.5.703
AB  - 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 \(\partial\Omega(t)\).
KW  - quasilinear parabolic equation
KW  - time-dependent domain
KW  - smoothing effects
SN  - 1232-9274
ER  -