Uniqueness and stability of regional blow-up in a porous-medium equation
Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 6, p. 927-960
@article{AIHPC_2002__19_6_927_0,
     author = {Cort\'azar, Carmen and Del Pino, Manuel and Elgueta, Manuel},
     title = {Uniqueness and stability of regional blow-up in a porous-medium equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {19},
     number = {6},
     year = {2002},
     pages = {927-960},
     zbl = {1018.35062},
     mrnumber = {1939091},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_6_927_0}
}
Cortázar, Carmen; del Pino, Manuel; Elgueta, Manuel. Uniqueness and stability of regional blow-up in a porous-medium equation. Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 6, pp. 927-960. http://www.numdam.org/item/AIHPC_2002__19_6_927_0/

[1] Cortázar C., Elgueta M., Felmer P., Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Comm. P.D.E. 21 (1996) 507-520. | MR 1387457 | Zbl 0854.35033

[2] Cortázar C., Elgueta M., Felmer P., Uniqueness of positive solutions of Δu+f(u)=0 in RN, N≥3, Arch. Rat. Mech. Anal. 142 (1998) 127-141. | Zbl 0912.35059

[3] Cortázar C., Elgueta M., Felmer P., On a semilinear elliptic problem in RN with a non-lipschitzian nonlinearity, Adv. Differential Equations 1 (2) (1996) 199-218. | MR 1364001 | Zbl 0845.35031

[4] Cortázar C., Del Pino M., Elgueta M., On the blow-up set for ut=Δum+um, m>1, Indiana Univ. Math. J. 47 (1998) 541-561. | Zbl 0916.35056

[5] Cortázar C., Del Pino M., Elgueta M., The problem of uniqueness of the limit in a semilinear heat equation, Comm. Partial Differential Equations 24 (1999) 2147-2172. | MR 1720758 | Zbl 0940.35107

[6] Feireisl E., Petzeltova H., Convergence to a ground state as threshold phenomenos in nonlinear parabolic equations, Differential Integral Equations 10 (1997) 181-196. | MR 1424805 | Zbl 0879.35023

[7] Feireisl E., Simondon F., Convergence for degenerate parabolic equations, J. Differential Equations 152 (2) (1999) 439-466. | MR 1674569 | Zbl 0928.35086

[8] E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, Preprint. | MR 1800136

[9] Fermanian Kammerer C., Merle F., Zaag H., Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Ann. 317 (2000) 347-387. | MR 1764243 | Zbl 0971.35038

[10] Fujita H., On the blowing-up of solutions of the Cauchy problem for ut=Δu+u1+α, J. Fac. Sci. Univ. Tokyo 13 (1966) 109-124. | Zbl 0163.34002

[11] Galaktionov V., On a blow-up set for the quasilinear heat equation ut=(uσux)x+uσ+1, J. Differential Equations 101 (1993) 66-79. | Zbl 0802.35065

[12] Galaktionov V., Blow-up for quasilinear heat equations with critical Fujita's exponent, Proc. Roy. Soc. Edinburgh 124A (1994) 517-525. | MR 1286917 | Zbl 0808.35053

[13] Galaktionov V., Peletier L.A., Asymptotic behaviour near finite-time extinction for the fast difussion equation, Arch. Rat. Mech. Anal. 139 (1997) 83-98. | MR 1475779 | Zbl 0885.35058

[14] Galaktionov V., Vazquez J.L., Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1) (1997) 1-67. | MR 1423231 | Zbl 0874.35057

[15] Giga Y., Kohn R., Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36 (1987) 1-40. | MR 876989 | Zbl 0601.35052

[16] Giga Y., Kohn R., Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989) 845-884. | MR 1003437 | Zbl 0703.35020

[17] Gui C., Symmetry of the blow-up set of a porous medium equation, Comm. Pure Appl. Math. 48 (1995) 471-500. | MR 1329829 | Zbl 0827.35014

[18] Hale J., Raugel G., Convergence in gradient-like and applications, Z. Angew. Math. Phys. 43 (1992) 63-124. | MR 1149371 | Zbl 0751.58033

[19] Haraux A., Polacik P., Convergence to a positive equilibrium for some nonlinear evolution equations in a ball, Acta Math. Univ. Comeniane 61 (1992) 129-141. | MR 1205867 | Zbl 0824.35011

[20] Korevaar N., Mazzeo R., Pacard F., Schoen R., Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Mat. 135 (2) (1999) 233-272. | MR 1666838 | Zbl 0958.53032

[21] Ladyzenskaja O.A., Solonnikov V.A., Ural'Ceva N.N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, 1968. | MR 241822 | Zbl 0174.15403

[22] Matano H., Nonincrease of the lap number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo 1A 29 (1982) 401-411. | MR 672070 | Zbl 0496.35011

[23] Ni W.-M., Takagi I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247-281. | MR 1219814 | Zbl 0796.35056

[24] Merle F., Zaag H., Stability of the blow-up profile for equations of the type ut=Δu+|u|p−1u, Duke Math. J. 86 (1997) 143-195. | Zbl 0872.35049

[25] Merle F., Zaag H., Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998) 139-196. | MR 1488298 | Zbl 0899.35044

[26] Polacik P., Rybakowski K.P., Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations 124 (1996) 472-494. | MR 1370152 | Zbl 0845.35054

[27] Samarskii A., Galaktionov V., Kurdyumov V., Mikhailov A., Blow-up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, 1987, in Russian.

[28] Simon L., Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. of Math. 118 (1983) 525-571. | MR 727703 | Zbl 0549.35071

[29] Velázquez J., Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 42 (1993) 445-476. | MR 1237055 | Zbl 0802.35073