| Title:
|
On an over-determined problem of free boundary of a degenerate parabolic equation (English) |
| Author:
|
Pan, Jiaqing |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
58 |
| Issue:
|
6 |
| Year:
|
2013 |
| Pages:
|
657-671 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\|u_{0}\|_{L^{2}(\mathbb {R}^{N})}\geq C_{0}$. (English) |
| Keyword:
|
inverse problem |
| Keyword:
|
parabolic equation |
| Keyword:
|
absorption |
| MSC:
|
35K10 |
| MSC:
|
35K65 |
| idZBL:
|
Zbl 06312920 |
| idMR:
|
MR3162753 |
| DOI:
|
10.1007/s10492-013-0033-3 |
| . |
| Date available:
|
2013-11-09T20:17:18Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143504 |
| . |
| Reference:
|
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