Localization for a porous medium type equation in high dimensions

Gui, Changfeng; Kang, Xiaosong

doi:10.1090/S0002-9947-04-03613-X

Localization for a porous medium type equation in high dimensions

Authors: Changfeng Gui and Xiaosong Kang
Journal: Trans. Amer. Math. Soc. 356 (2004), 4273-4285
MSC (2000): Primary 35K15, 35K55, 35K65; Secondary 35J40
DOI: https://doi.org/10.1090/S0002-9947-04-03613-X
Published electronically: May 28, 2004
MathSciNet review: 2067119
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Abstract: We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla(u^ {\sigma} \nabla u)+u^ \beta$ , $x \in \mathbf{R}^ N$ , , $\beta >\sigma +1$ , $\sigma>0,$ in the case of arbitrary compactly supported initial functions . We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname{supp}u_0$ and the blow-up time . Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.

1. D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, https://doi.org/10.1090/S0002-9947-1983-0712265-1
2. F. V. Bunkin, V. A. Galaktionov, N. A. Kirichenko, S. P. Kurdyumov, and A. A. Samarskiĭ, A nonlinear boundary value problem of ignition by radiation, Zh. Vychisl. Mat. i Mat. Fiz. 28 (1988), no. 4, 549–559, 623 (Russian); English transl., U.S.S.R. Comput. Math. and Math. Phys. 28 (1988), no. 2, 157–164 (1989). MR 943615, https://doi.org/10.1016/0041-5553(88)90161-9
3. Carmen Cortázar, Manuel del Pino, and Manuel Elgueta, On the blow-up set for 𝑢_{𝑡}=Δ𝑢^{𝑚}+𝑢^{𝑚}, 𝑚>1, Indiana Univ. Math. J. 47 (1998), no. 2, 541–561. MR 1647932, https://doi.org/10.1512/iumj.1998.47.1399
4. Avner Friedman and Bryce McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR 783924, https://doi.org/10.1512/iumj.1985.34.34025
5. V. A. Galaktionov, A proof of the localization of unbounded solutions of the nonlinear parabolic equation 𝑢_{𝑡}=(𝑢^{𝜎}𝑢ₓ)ₓ+𝑢^{𝛽}, Differentsial′nye Uravneniya 21 (1985), no. 1, 15–23, 179–180 (Russian). MR 777775
6. V. A. Galaktionov, Asymptotic behavior of unbounded solutions of the nonlinear equation 𝑢_{𝑡}=(𝑢^{𝜎}𝑢ₓ)ₓ+𝑢^{𝛽} near a “singular” point, Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1293–1297 (Russian). MR 852454
7. Victor A. Galaktionov and Juan L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 399–433. Current developments in partial differential equations (Temuco, 1999). MR 1897690, https://doi.org/10.3934/dcds.2002.8.399
8. Victor A. Galaktionov and Juan L. Vazquez, Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations 127 (1996), no. 1, 1–40. MR 1387255, https://doi.org/10.1006/jdeq.1996.0059
9. Victor A. Galaktionov and Juan L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), no. 1, 1–67. MR 1423231, https://doi.org/10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R
10. Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, https://doi.org/10.1002/cpa.3160380304
11. Changfeng Gui, Symmetry of the blow-up set of a porous medium type equation, Comm. Pure Appl. Math. 48 (1995), no. 5, 471–500. MR 1329829, https://doi.org/10.1002/cpa.3160480502
12. Changfeng Gui, Wei-Ming Ni, and Xuefeng Wang, Further study on a nonlinear heat equation, J. Differential Equations 169 (2001), no. 2, 588–613. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 4 (Atlanta, GA/Lisbon, 1998). MR 1808476, https://doi.org/10.1006/jdeq.2000.3909
13. Changfeng Gui, Wei-Ming Ni, and Xuefeng Wang, On the stability and instability of positive steady states of a semilinear heat equation in 𝑅ⁿ, Comm. Pure Appl. Math. 45 (1992), no. 9, 1153–1181. MR 1177480, https://doi.org/10.1002/cpa.3160450906
14. A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 135–176, 287 (Russian). MR 898624
15. Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, https://doi.org/10.1137/1032046
16. Alexander A. Samarskii, Victor A. Galaktionov, Sergei P. Kurdyumov, and Alexander P. Mikhailov, Blow-up in quasilinear parabolic equations, De Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1995. Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. MR 1330922
17. Michel C. Delfour and Gert Sabidussi (eds.), Shape optimization and free boundaries, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 380, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1260971
18. J. J. L. Velázquez, Estimates on the (𝑛-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), no. 2, 445–476. MR 1237055, https://doi.org/10.1512/iumj.1993.42.42021
19. Xuefeng Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), no. 2, 549–590. MR 1153016, https://doi.org/10.1090/S0002-9947-1993-1153016-5
20. Fred B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), no. 2, 204–224. MR 764124, https://doi.org/10.1016/0022-0396(84)90081-0