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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 $\|u_{0}\|_{L^{2}(\mathbb {R}^{N})}\geq C_{0}$.
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65. A Series of Monographs and Textbooks. Academic Press New York (1975). MR 0450957 | Zbl 0314.46030
[2] Barenblatt, G. I.: On some unsteady motions and a liquid and gas in a porous medium. Prikl. Mat. Mekh. 16 (1952), 67-78 Russian. MR 0046217
[3] Esteban, J. R., Rodríguez, A., Vázquez, J. L.: A nonlinear heat equation with singular diffusivity. Commun. Partial Differ. Equations 13 (1988), 985-1039. DOI 10.1080/03605308808820566 | MR 0944437 | Zbl 0686.35066
[4] Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall Englewood Cliffs (1964). MR 0181836 | Zbl 0144.34903
[5] Gilding, B. H.: Properties of solutions of an equation in the theory of infiltration. Arch. Ration. Mech. Anal. 65 (1977), 203-225. DOI 10.1007/BF00280441 | MR 0447847 | Zbl 0366.76074
[6] Gilding, B. H., Peletier, L. A.: The Cauchy problem for an equation in the theory of infiltration. Arch. Ration. Mech. Anal. 61 (1976), 127-140. DOI 10.1007/BF00249701 | MR 0408428 | Zbl 0336.76037
[7] Kamynin, V. L.: On the inverse problem of determining the leading coefficient in parabolic equations. Math. Notes 84 (2008), 45-54. DOI 10.1134/S0001434608070043 | MR 2451884 | Zbl 1219.35361
[8] Kamynin, V. L.: On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition. Math. Notes 77 (2005), 482-493. DOI 10.1007/s11006-005-0047-6 | MR 2178019 | Zbl 1075.35106
[9] Kozhanov, A. I.: Solvability of the inverse problem of finding thermal conductivity. Sib. Math. J. 46 (2005), 841-856. DOI 10.1007/s11202-005-0082-2 | MR 2187462 | Zbl 1119.35117
[10] Ladyzhenskaja, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs Vol. 23. American Mathematical Society Providence (1968) Izdat. Nauka, Moskva, 1967 Russian. DOI 10.1090/mmono/023 | MR 0241822
[11] Peletier, L. A.: On the existence of an interface in nonlinear diffusion processes. Ordinary and Partial Differential Equations. Proceedings of the conference held at Dundee, Scotland, 26-29 March, 1974. Lecture Notes in Mathematics 415 B. D. Sleeman et al. Springer Berlin (1974), 412-416. DOI 10.1007/BFb0065558 | MR 0422867 | Zbl 0309.35041
[12] Vazquez, J. L.: An introduction to the mathematical theory of the porous medium equation. Shape Optimization and Free Boundaries. Proceedings of the NATO Advanced Study Institute and Séminaire de mathématiques supérieures, held Montréal, Canada, June 25--July 13, 1990. NATO ASI Series. Series C. Mathematical and Physical Sciences 380 M. C. Delfour et al. Kluwer Academic Publishers Dordrecht (1992), 347-389. MR 1260981 | Zbl 0765.76086
[13] Wu, Z., Yin, J., Wang, C.: Elliptic and parabolic equations. World Scientific Hackensack (2006). MR 2309679 | Zbl 1108.35001
[14] Yu, W.: Well-posedness of a parabolic inverse problem. Acta Math. Appl. Sin., Engl. Ser. 13 (1997), 329-336. DOI 10.1007/BF02025888 | MR 1473818 | Zbl 0929.35178
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