We investigate stationary solutions and asymptotic behaviour of solutions of two boundary value problems for semilinear parabolic equations. These equations involve both blow up and damping terms and they were studied by several authors. Our main goal is to fill some gaps in these studies.
@article{118561, author = {Pavol Quittner}, title = {On global existence and stationary solutions for two classes of semilinear parabolic problems}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {34}, year = {1993}, pages = {105-124}, zbl = {0794.35089}, mrnumber = {1240209}, language = {en}, url = {http://dml.mathdoc.fr/item/118561} }
Quittner, Pavol. On global existence and stationary solutions for two classes of semilinear parabolic problems. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 105-124. http://gdmltest.u-ga.fr/item/118561/
Blow up in $\Bbb R^n$ for a parabolic equation with a damping nonlinear gradient term, Nonlinear Diffusion Equations and Their Equilibrium States, 3, N.G. Lloyd, W.M. Ni, L.A. Peletier, J. Serrin (eds.), Birkhäuser, Boston-Basel-Berlin, 1992. | MR 1167826
Liapunov operators and stabilization in strongly order preserving dynamical systems, Diff. Integral Equs. 4 (1991), 15-24. (1991) | MR 1079608
Parabolic evolution equations and nonlinear boundary conditions, J. Diff. Equations 72 (1988), 201-269. (1988) | MR 0932367 | Zbl 0658.34011
Dynamic theory of quasilinear parabolic equations: II. Reaction-diffusion systems, Diff. Integral Equs. 3 (1990), 13-75. (1990) | MR 1014726 | Zbl 0729.35062
On a class of superlinear elliptic problems, Comm. in P.D.E. 2 (1977), 601-614. (1977) | MR 0509489
On a class of nonlinear elliptic equations, Proceedings of the Banach Center, to appear. | MR 1205813 | Zbl 0819.35051
Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenianae 60 (1991), 35-103. (1991) | MR 1120596 | Zbl 0743.35038
Some blow up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal. 20 (1989), 886-907. (1989) | MR 1000727
On the elliptic problem $\triangle u-|\nabla u|^q+łu^p=0$, Nonlinear Diffusion Equations and Their Equilibrium States, Vol. I, W.-M. Ni, L.A. Peletier, J.B. Serrin (eds.), Springer, New York, 1988, pp. 237-243. | MR 0956067 | Zbl 0699.35102
Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann. 284 (1989), 285-305. (1989) | MR 1000112 | Zbl 0652.35065
A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. pures et appl. 61 (1982), 41-63. (1982) | MR 0664341 | Zbl 0452.35030
Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc. 111 (1991), 795-801. (1991) | MR 1052569 | Zbl 0768.35047
The blow-up rate for the heat equation with a nonlinear boundary condition, Math. Meth. Appl. Sci. 14 (1991), 197-205. (1991) | MR 1099325 | Zbl 0735.35014
Radial ground states for a semilinear elliptic equation with a gradient term, to appear in Adv. Math. Sci. Appl. | MR 1239247
Local existence of general nonlinear parabolic systems, to appear.
Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989), 207-217. (1989) | MR 1013085 | Zbl 0661.35053
Elliptic partial differential equations of second order, second edition, Springer, Berlin-Heidelberg-New York-Tokyo, 1983. | MR 0737190 | Zbl 1042.35002
Global behavior of positive solutions to a semilinear equation with a nonlinear flux condition, IMA Preprint Series #810, May 1991. | MR 1120912
Blow-up for semilinear parabolic equations with a gradient term, Math. Meth. Appl. Sci. 14 (1991), 413-417. (1991) | MR 1119238 | Zbl 0768.35049
On positive solutions of semilinear elliptic problems, Comment. Math. Univ. Carolinae 30 (1989), 579-585. (1989) | MR 1031874 | Zbl 0698.35057
Global Solution Branches of Two Point Boundary Value Problems, LNM 1458, Springer, Berlin-Heidelberg, 1990. | MR 1090827 | Zbl 0780.34010
Existence and non-existence results for ground states of quasilinear elliptic equations, to appear. | Zbl 0795.35027
Introduction to Fourier Analysis on Euclidean Spaces, Princeton, New Jersey, 1971, . | MR 0304972
Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. Henri Poincaré, Analyse non linéaire 5 (1988), 425-464. (1988) | MR 0970849 | Zbl 0664.35022
Plateau's Problem and the Calculus of Variations, Mathematical Notes 35, Princeton, New Jersey, 1988. | MR 0992402 | Zbl 0694.49028
Variational methods and their applications to nonlinear partial differential equations and Hamiltonian systems, Springer, Berlin-Heidelberg, 1990. | MR 1078018