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Vanishing latent heat limit in a Stefan-like problem arising in biology. (English) Zbl 1049.92035

Summary: We consider a two phase Stefan problem with a reaction term in arbitrary space dimension and prove that as the latent heat coefficient tends to zero, its weak solution converges to the weak solution of the corresponding problem with zero latent heat, which is obtained as the spatial segregation limit of a competition-diffusion system. In particular, we obtain a uniform convergence result for the corresponding interfaces in the one-dimensional case.

MSC:

92D40 Ecology
35K57 Reaction-diffusion equations
35R35 Free boundary problems for PDEs
35K65 Degenerate parabolic equations
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[1] Aronson, D.G.; Serrin, J., Local behavior of solutions of quasilinear parabolic equations, Arch. rational mech. anal., 25, 81-122, (1967) · Zbl 0154.12001
[2] Dancer, E.N.; Hilhorst, D.; Mimura, M.; Peletier, L.A., Spatial segregation limit of a competition-diffusion system, European J. appl. math., 10, 97-115, (1999) · Zbl 0982.92031
[3] DiBenedetto, E.; Friedman, A., The ill-posed hele – show model and the Stefan problem for supercooled water, Trans. amer. math. soc., 282, 183-204, (1984) · Zbl 0621.35102
[4] Elliott, C.M.; Schätzle, R., The limit of the anisotropic double-obstacle allen – cahn equation, Proc. roy. soc. Edinburgh, 126A, 1217-1234, (1996) · Zbl 0865.35073
[5] Evans, L.C., A chemical diffusion – reaction free boundary problem, Nonlinear anal., 6, 5, 455-466, (1982) · Zbl 0511.35082
[6] Kishimoto, K.; Weinberger, H., The spatial homogeneity of stable equiliblia of some reaction – diffusion system on convex domain, J. differential equations, 58, 15-21, (1985) · Zbl 0599.35080
[7] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI (Translation of Mathematical Monographs, Vol. 23, 1968).
[8] Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. fac. sci. univ. Tokyo IA, 29, 2, 401-441, (1982) · Zbl 0496.35011
[9] Mimura, M.; Yamada, Y.; Yotsutani, S., A free boundary problem in ecology, Japan J. appl. math., 2, 151-186, (1985) · Zbl 0593.92019
[10] Mimura, M.; Yamada, Y.; Yotsutani, S., Stability analysis for free boundary problems in ecology, Hiroshima math. J., 16, 3, 477-498, (1986) · Zbl 0617.35135
[11] Mimura, M.; Yamada, Y.; Yotsutani, S., Free boundary problems for some reaction – diffusion equations, Hiroshima math. J., 17, 2, 241-280, (1987) · Zbl 0649.35089
[12] Simon, J., Compact sets in the space Lp(0,T;B), Ann. math. pura appl. S.4, 146, 65-96, (1987) · Zbl 0629.46031
[13] Tarzia, D.A., Sur le problème de Stefan à deux phases, C. R. acad. sci. Paris, 288 A, 941-944, (1979) · Zbl 0416.35001
[14] Tarzia, D.A., Etude de l’inéquation variationelle proposée par duvaut pour le problème de Stefan à deux phases, I, Boll. un. mat. ital., 1-B, 865-883, (1982) · Zbl 0504.35081
[15] Tarzia, D.A., Etude de l’inéquation variationelle proposée par duvaut pour le problème de Stefan à deux phases, II, Boll. un. mat. ital., 2-B, 589-603, (1983) · Zbl 0526.35046
[16] Trudinger, N.S., Pointwise estimates and quasilinear parabolic equations, Comm. pure appl. math., 21, 205-226, (1968) · Zbl 0159.39303
[17] Yotsutani, S., Stefan problems with the unilateral boundary condition on the fixed boundary II, Osaka J. math., 20, 803-844, (1983) · Zbl 0579.35085
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