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
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by Changfeng Gui and Xiaosong Kang PDF

Trans. Amer. Math. Soc. 356 (2004), 4273-4285 Request permission

Abstract:

We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla (u^ {\sigma } \nabla u)+u^ \beta$, $x \in \mathbf {R}^ N$, $N>1$, $\beta >\sigma +1$, $\sigma >0,$ in the case of arbitrary compactly supported initial functions $u_0$. 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 $T$. Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.

References

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, DOI 10.1090/S0002-9947-1983-0712265-1
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, DOI 10.1016/0041-5553(88)90161-9
Carmen Cortázar, Manuel del Pino, and Manuel Elgueta, On the blow-up set for $u_t=\Delta u^m+u^m$, $m>1$, Indiana Univ. Math. J. 47 (1998), no. 2, 541–561. MR 1647932, DOI 10.1512/iumj.1998.47.1399
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, DOI 10.1512/iumj.1985.34.34025
V. A. Galaktionov, A proof of the localization of unbounded solutions of the nonlinear parabolic equation $u_t=(u^\sigma u_x)_x+u^\beta$, Differentsial′nye Uravneniya 21 (1985), no. 1, 15–23, 179–180 (Russian). MR 777775
V. A. Galaktionov, Asymptotic behavior of unbounded solutions of the nonlinear equation $u_t=(u^\sigma u_x)_x+u^\beta$ near a “singular” point, Dokl. Akad. Nauk SSSR 288 (1986), no. 6, 1293–1297 (Russian). MR 852454
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, DOI 10.3934/dcds.2002.8.399
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, DOI 10.1006/jdeq.1996.0059
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, DOI 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R
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, DOI 10.1002/cpa.3160380304
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, DOI 10.1002/cpa.3160480502
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, DOI 10.1006/jdeq.2000.3909
Changfeng Gui, Wei-Ming Ni, and Xuefeng Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\textbf {R}^n$, Comm. Pure Appl. Math. 45 (1992), no. 9, 1153–1181. MR 1177480, DOI 10.1002/cpa.3160450906
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
Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI 10.1137/1032046
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, DOI 10.1515/9783110889864.535
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, DOI 10.1007/978-94-011-2710-3
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, DOI 10.1512/iumj.1993.42.42021
Xuefeng Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), no. 2, 549–590. MR 1153016, DOI 10.1090/S0002-9947-1993-1153016-5
Fred B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), no. 2, 204–224. MR 764124, DOI 10.1016/0022-0396(84)90081-0