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global existence; blow up; semilinear parabolic equation; stationary solution
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.
[AW] Alfonsi L., Weissler F.B.: 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
[AH] Alikakos N.D., Hess P.: Liapunov operators and stabilization in strongly order preserving dynamical systems. Diff. Integral Equs. 4 (1991), 15-24. MR 1079608
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[C] Chipot M.: On a class of nonlinear elliptic equations. Proceedings of the Banach Center, to appear. MR 1205813 | Zbl 0819.35051
[CFQ] Chipot M., Fila M., Quittner P.: 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. MR 1120596 | Zbl 0743.35038
[CW1] Chipot M., Weissler F.B.: Some blow up results for a nonlinear parabolic equation with a gradient term. SIAM J. Math. Anal. 20 (1989), 886-907. MR 1000727
[CW2] Chipot M., Weissler F.B.: 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
[E] Escher J.: Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions. Math. Ann. 284 (1989), 285-305. MR 1000112 | Zbl 0652.35065
[FLN] de Figueiredo D.G., Lions P.-L., Nussbaum R.D.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. pures et appl. 61 (1982), 41-63. MR 0664341 | Zbl 0452.35030
[F] Fila M.: Remarks on blow up for a nonlinear parabolic equation with a gradient term. Proc. Amer. Math. Soc. 111 (1991), 795-801. MR 1052569 | Zbl 0768.35047
[FQ1] Fila M., Quittner P.: The blow-up rate for the heat equation with a nonlinear boundary condition. Math. Meth. Appl. Sci. 14 (1991), 197-205. MR 1099325 | Zbl 0735.35014
[FQ2] Fila M., Quittner P.: Radial ground states for a semilinear elliptic equation with a gradient term. to appear in Adv. Math. Sci. Appl. MR 1239247
[FK] Filo J., Kačur J.: Local existence of general nonlinear parabolic systems. to appear.
[KP] Kawohl B., Peletier L.A.: Observations on blow up and dead cores for nonlinear parabolic equations. Math. Z. 202 (1989), 207-217. MR 1013085 | Zbl 0661.35053
[GT] Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. second edition, Springer, Berlin-Heidelberg-New York-Tokyo, 1983. MR 0737190 | Zbl 1042.35002
[LGMW] López Gómez J., Márquez V., Wolanski N.: Global behavior of positive solutions to a semilinear equation with a nonlinear flux condition. IMA Preprint Series #810, May 1991. MR 1120912
[Q1] Quittner P.: Blow-up for semilinear parabolic equations with a gradient term. Math. Meth. Appl. Sci. 14 (1991), 413-417. MR 1119238 | Zbl 0768.35049
[Q2] Quittner P.: On positive solutions of semilinear elliptic problems. Comment. Math. Univ. Carolinae 30 (1989), 579-585. MR 1031874 | Zbl 0698.35057
[S] Schaaf R.: Global Solution Branches of Two Point Boundary Value Problems. LNM 1458, Springer, Berlin-Heidelberg, 1990. MR 1090827 | Zbl 0780.34010
[SZ] Serrin J.B., Zou H.: Existence and non-existence results for ground states of quasilinear elliptic equations. to appear. Zbl 0795.35027
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[S1] Struwe M.: Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces. Ann. Inst. Henri Poincaré, Analyse non linéaire 5 (1988), 425-464. MR 0970849 | Zbl 0664.35022
[S2] Struwe M.: Plateau's Problem and the Calculus of Variations. Mathematical Notes 35, Princeton, New Jersey, 1988. MR 0992402 | Zbl 0694.49028
[S3] Struwe M.: Variational methods and their applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin-Heidelberg, 1990. MR 1078018
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