Abstract
In this paper, we consider a localized problem with free boundary for the heat equation in higher space dimensions and heterogeneous environment. For simplicity, we assume that the environment and solution are radially symmetric. First, by using the contraction mapping theorem, we prove that the local solution exists and is unique. Then, some sufficient conditions are given under which the solution will blow up in finite time. Our results indicate that the blowup occurs if the initial data are sufficiently large. Finally, the long time behavior of the global solution is discussed. It is shown that the global fast solution does exist if the initial data are sufficiently small, while the global slow solution is possible if the initial data are suitably large.
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Bimpong-Bota K, Ortoleva P, Ross J. Far-from-equilibrium phenomena at local sites of reaction. J Chem Phys, 1974, 60: 3124–3133
Caffarelli L, Salsa S. A Geometric Approach to Free Boundary Problems. In: Graduate Studies in Mathematics, 68. Providence, RI: Amer Math Soc, 2005
Chadam J M, Peirce A, Yin H M. The blowup property of solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl, 1992, 169: 313–328
Chan C Y, Yang J. Complete blow-up for degenerate semilinear parabolic equations. J Comp Appl Math, 2000, 113: 353–364
Chen X F, Friedman A. A free boundary problem arising in a model of wound healing. SIAM J Math Anal, 2000, 32: 778–800
Crank J. Free and Moving Boundary Problem. Oxford: Clarendon Press, 1984
Du Y H, Lin Z G. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal, 2010, 42: 377–405
Du Y H, Guo Z M. Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary II. J Differential Equations, 2011, 250: 4336–4366
Fasano A, Primicerio M. Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions. J Math Anal Appl, 1979, 72: 247–273
Fila M, Souplet P. Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Interface Free Boundary, 2001, 3: 337–344
Gan Z H, Guo B L. Blow-up phenomena of the vector nonlinear Schrödinger equations with magnetic fields. Sci China Math, 2011, 54: 2111–2122
Ghidouche H, Souplet P, Tarzia D. Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary. Proc Amer Math Soc, 2001, 129: 781–792
Hilhorst D, Mimura M, Schatzle R. Vanishing latent heat limit in a Stefan-like problem arising in biology. Nonlinear Anal Real World Appl, 2003, 4: 261–285
Kim K I, Lin Z G, Ling Z. Global existence and blowup of solutions to a free boundary problem for mutualistic model. Sci China Math, 2010, 53: 2085–2095
Ladyzenskaja O A, Solonnikov V A, Ural’ceva N N. Linear and Quasilinear Equations of Parabolic Type. Providence, RI: Amer Math Soc, 1968
Lin Z G. A free boundary problem for a predator-prey model. Nonlinearity, 2007, 20: 1883–1892
Mimura M, Yamada Y, Yotsutani S. Free boundary problems for some reaction-diffusion equations. Hiroshima Math J, 1987, 17: 241–280
Ogawa M, Tani A A. Incompressible perfect fluid motion with free boundary of finite depth. Adv Math Sci Appl, 2003, 13: 201–223
Ortoleva P, Ross J. Local structures in chemical reactions with heterogeneous catalysis. J Chem Phys, 1972, 56: 4397–4452
Ricci R, Tarzia D A. Asymptotic behavior of the solutions of the dead-core problem. Nonlinear Anal, 1989, 13: 405–411
Rubinsky B, Shitzer A. Analysis of a Stefan-like problem in a biological tissue around a cryosurgical probe. J Heat Transfer, 1975, 98: 514–519
Rubinstein L I. The Stefan Problem. Providence, RI: Amer Math Soc, 1971
Souplet P. Stability and continuous dependence of solutions to one-phase Stefan problems for semilinear parabolic equations. Port Math, 2002, 59: 315–323
Sun Y Z, Zhang Z F. A blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations. Sci China Math, 2011, 54: 105–116
Zhang P, Zhang Z F. On the local wellposedness of 3-D water wave problem with vorticity. Sci China Ser A, 2007, 50: 1065–1077
Zhou P, Bao J, Lin Z G. Global existence and blowup of a localized problem with free boundary. Nonlinear Anal, 2011, 74: 2523–2533
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Zhou, P., Lin, Z. Global fast and slow solutions of a localized problem with free boundary. Sci. China Math. 55, 1937–1950 (2012). https://doi.org/10.1007/s11425-012-4443-6
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DOI: https://doi.org/10.1007/s11425-012-4443-6