Previous |  Up |  Next

Article

Keywords:
a priori estimate; global existence; parabolic equation; superlinear nonlinearity
Summary:
In this survey we consider superlinear parabolic problems which possess both blowing-up and global solutions and we study a priori estimates of global solutions.
References:
[1] H. Brezis, R. E. L. Turner: On a class of superlinear elliptic problems. Commun. Partial Differ. Equations 2 (1977), 601–614. MR 0509489
[2] T. Cazenave, P.-L. Lions: Solutions globales d’équations de la chaleur semi linéaires. Commun. Partial Differ. Equations 9 (1984), 955–978. MR 0755928
[3] Ph. Clément, D. G. de Figueiredo, E. Mitidieri: A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities. Nonlinear partial differential equations, A. Benkirane at al (eds.), Pitman Research Notes in Math. 343, Harlow, Longman, 1996, pp. 73–91. MR 1417272
[4] M. Escobedo, M. A. Herrero: Boundedness and blow up for a semilinear reaction-diffusion system. J. Differ. Equations 89 (1991), 176–202. MR 1088342
[5] M. Fila, P. Souplet, F. Weissler: Linear and nonlinear heat equations in $L^p_\delta $ spaces and universal bounds for global solutions. Preprint. MR 1835063
[6] V. Galaktionov, J. L. Vázquez: Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Applied Math. 50 (1997), 1–67. MR 1423231
[7] B. Gidas, J. Spruck: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equations 6 (1991), 883–901. MR 0619749
[8] Y. Giga: A bound for global solutions of semilinear heat equations. Commun. Math. Phys. 103 (1986), 415–421. MR 0832917 | Zbl 0595.35057
[9] Y. Giga, R. V. Kohn: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36 (1987), 1–40. MR 0876989
[10] Y. Gu, M. Wang: Existence of positive stationary solutions and threshold results for a reaction-diffusion system. J. Differ. Equations 130 (1996), 277–291. MR 1410888
[11] B. Hu: Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition. Differ. Integral Equations 9 (1996), 891–901. MR 1392086
[12] S. Kaplan: On the growth of solutions of quasi-linear parabolic equations. Commun. Pure Appl. Math. 16 (1963), 305–330. MR 0160044 | Zbl 0156.33503
[13] H. A. Levine: A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations. Z. Angew. Math. Phys. 42 (1992), 408–430. MR 1115199
[14] W.-M. Ni, P. E. Sacks, J. Tavantzis: On the asymptotic behavior of solutions of certain quasilinear parabolic equations. J. Differ. Equations 54 (1984), 97–120. MR 0756548
[15] P. Quittner: A priori bounds for global solutions of a semilinear parabolic problem. Acta Math. Univ. Comenianae 68 (1999), 195–203. MR 1757788 | Zbl 0940.35112
[16] P. Quittner: Universal bound for global positive solutions of a superlinear parabolic problem. Preprint. MR 1839765 | Zbl 0981.35010
[17] P. Quittner: Signed solutions for a semilinear elliptic problem. Differ. Integral Equations 11 (1998), 551–559. MR 1666269 | Zbl 1131.35339
[18] P. Quittner: A priori estimates of global solutions and multiple equilibria of a parabolic problem involving measure. Preprint.
[19] P. Quittner: Transition from decay to blow-up in a parabolic system. Arch. Math. (Brno) 34 (1998), 199–206. MR 1629705 | Zbl 0911.35062
[20] P. Quittner, Ph. Souplet: In preparation.
[21] H. Zou: Existence of positive solutions of semilinear elliptic systems without variational structure. Preprint.
Partner of
EuDML logo