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

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 $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$, where H _u(T ₀) = {x, ℝ^N: u(x, T ₀) > 0}. In this paper, we first establish a priori estimate u _t ⩾ C(t)u and a more precise Poincaré 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 C ₀ and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0$.