On an over-determined problem of free boundary of a degenerate parabolic equation
| Autor: | Jiaqing Pan |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Applications of Mathematics. 58:657-671 |
| ISSN: | 1572-9109 0862-7940 |
| DOI: | 10.1007/s10492-013-0033-3 |
| Popis: | 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 T0, to decide the initial value u0 such that the solution u(x, t) satisfies \(\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K\), where Hu(T0) = {x, ℝN: u(x, T0) > 0}. In this paper, we first establish a priori estimate ut ⩾ C(t)u and a more precise Poincare type inequality \(\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 \), and then, we give a positive constant C0 and assert the main results are true if only \(\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 \). |
| Databáze: | OpenAIRE |
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