On an over-determined problem of free boundary of a degenerate parabolic equation

Pan, Jiaqing. "On an over-determined problem of free boundary of a degenerate parabolic equation." Applications of Mathematics 58.6 (2013): 657-671. <http://eudml.org/doc/260642>.

@article{Pan2013,
abstract = {This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants $K$ and $T_\{0\}$, to decide the initial value $u_\{0\}$ such that the solution $u(x,t)$ satisfies $\sup _\{x\in H_\{u\}(T_\{0\})\}|x|\ge K$, where $H_\{u\}(T_\{0\})=\lbrace x\in \mathbb \{R\}^\{N\}\colon u(x,T_\{0\})>0\rbrace $. In this paper, we first establish a priori estimate $u_\{t\}\ge C(t)u$ and a more precise Poincaré type inequality $\Vert \phi \Vert ^\{2\}_\{L^\{2\}(B_\{\rho \})\}\le \rho \Vert \nabla \phi \Vert ^\{2\}_\{L^\{2\}(B_\{\rho \})\}$, and then, we give a positive constant $C_\{0\}$ and assert the main results are true if only $\Vert u_\{0\}\Vert _\{L^\{2\}(\mathbb \{R\}^\{N\})\}\ge C_\{0\}$.},
author = {Pan, Jiaqing},
journal = {Applications of Mathematics},
keywords = {inverse problem; parabolic equation; absorption; degenerate parabolic equation; finite speed of propagation; inverse problem},
language = {eng},
number = {6},
pages = {657-671},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On an over-determined problem of free boundary of a degenerate parabolic equation},
url = {http://eudml.org/doc/260642},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Pan, Jiaqing
TI - On an over-determined problem of free boundary of a degenerate parabolic equation
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 6
SP - 657
EP - 671
AB - This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants $K$ and $T_{0}$, to decide the initial value $u_{0}$ such that the solution $u(x,t)$ satisfies $\sup _{x\in H_{u}(T_{0})}|x|\ge K$, where $H_{u}(T_{0})=\lbrace x\in \mathbb {R}^{N}\colon u(x,T_{0})>0\rbrace $. In this paper, we first establish a priori estimate $u_{t}\ge C(t)u$ and a more precise Poincaré type inequality $\Vert \phi \Vert ^{2}_{L^{2}(B_{\rho })}\le \rho \Vert \nabla \phi \Vert ^{2}_{L^{2}(B_{\rho })}$, and then, we give a positive constant $C_{0}$ and assert the main results are true if only $\Vert u_{0}\Vert _{L^{2}(\mathbb {R}^{N})}\ge C_{0}$.
LA - eng
KW - inverse problem; parabolic equation; absorption; degenerate parabolic equation; finite speed of propagation; inverse problem
UR - http://eudml.org/doc/260642
ER -