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inverse problem; parabolic equation; absorption
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 $\|u_{0}\|_{L^{2}(\mathbb {R}^{N})}\geq C_{0}$.
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