On an over-determined problem of free boundary of a degenerate parabolic equation
Applications of Mathematics, Tome 58 (2013) no. 6, pp. 657-671
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 $\sup _{x\in H_{u}(T_{0})}|x|\geq K$, where $H_{u}(T_{0})=\{x\in \mathbb {R}^{N}\colon u(x,T_{0})>0\}$. In this paper, we first establish a priori estimate $u_{t}\geq C(t)u$ and a more precise Poincaré type inequality $\|\phi \|^{2}_{L^{2}(B_{\rho })}\leq \rho \|\nabla \phi \|^{2}_{L^{2}(B_{\rho })}$, and then, we give a positive constant $C_{0}$ and assert the main results are true if only $\|u_{0}\|_{L^{2}(\mathbb {R}^{N})}\geq C_{0}$.
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 $\sup _{x\in H_{u}(T_{0})}|x|\geq K$, where $H_{u}(T_{0})=\{x\in \mathbb {R}^{N}\colon u(x,T_{0})>0\}$. In this paper, we first establish a priori estimate $u_{t}\geq C(t)u$ and a more precise Poincaré type inequality $\|\phi \|^{2}_{L^{2}(B_{\rho })}\leq \rho \|\nabla \phi \|^{2}_{L^{2}(B_{\rho })}$, and then, we give a positive constant $C_{0}$ and assert the main results are true if only $\|u_{0}\|_{L^{2}(\mathbb {R}^{N})}\geq C_{0}$.
DOI :
10.1007/s10492-013-0033-3
Classification :
35K10, 35K65
Keywords: inverse problem; parabolic equation; absorption
Keywords: inverse problem; parabolic equation; absorption
@article{10_1007_s10492_013_0033_3,
author = {Pan, Jiaqing},
title = {On an over-determined problem of free boundary of a degenerate parabolic equation},
journal = {Applications of Mathematics},
pages = {657--671},
year = {2013},
volume = {58},
number = {6},
doi = {10.1007/s10492-013-0033-3},
mrnumber = {3162753},
zbl = {06312920},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0033-3/}
}
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 SP - 657 EP - 671 VL - 58 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0033-3/ DO - 10.1007/s10492-013-0033-3 LA - en ID - 10_1007_s10492_013_0033_3 ER -
%0 Journal Article %A Pan, Jiaqing %T On an over-determined problem of free boundary of a degenerate parabolic equation %J Applications of Mathematics %D 2013 %P 657-671 %V 58 %N 6 %U http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0033-3/ %R 10.1007/s10492-013-0033-3 %G en %F 10_1007_s10492_013_0033_3
Pan, Jiaqing. On an over-determined problem of free boundary of a degenerate parabolic equation. Applications of Mathematics, Tome 58 (2013) no. 6, pp. 657-671. doi: 10.1007/s10492-013-0033-3
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