# zbMATH — the first resource for mathematics

On the asymptotic behavior of solutions of certain quasilinear parabolic equations. (English) Zbl 0565.35053
The large time behavior of non negative weak solutions of nonlinear parabolic problems is investigated by the authors. Conditions are given under which global $$L^ 1$$ solutions are unbounded or uniformly bounded away from the origin. Next their stabilization towards $$+\infty$$, 0 or unstable equilibrium is analyzed.
Reviewer: M.Langlais

##### MSC:
 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
Full Text:
##### References:
 [1] Aronson, D.G, On the Green’s function for second order parabolic differential equations with discontinuous coefficients, Bull. amer. math. soc., 69, 841-847, (1963) · Zbl 0154.11903 [2] {\scP. Benilan, M. G. Crandall, and M. Pierre}, “Solutions of the Porous Medium Equation in $$R$$^{N} under Optimal Conditions on the Initial Values,” T.S.R. 2387, Mathematics Research Center, Madison, Wisconsin. · Zbl 0552.35045 [3] Brézis, H; Nirenberg, L, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029 [4] Crandall, M.G, An introduction to evolution governed by accretive operators, () · Zbl 0339.35049 [5] Dafermos, C.M, Asymptotic behavior of solutions of evolution equations, () · Zbl 0499.35015 [6] 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 appl., 61, 41-63, (1982) · Zbl 0452.35030 [7] Evans, L.C, Application of nonlinear semigroup theory to certain partial differential equations, () · Zbl 0471.35039 [8] Friedman, A, Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, N. J · Zbl 0144.34903 [9] Gidas, B; Ni, W.-M; Nirenberg, L, Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020 [10] Gilbarg, D; Trudinger, N, Elliptic partial differential equations of second order, (1977), Springer-Verlag New York/Berlin · Zbl 0361.35003 [11] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001 [12] Holland, C, Limiting behavior of a class of nonlinear reaction diffusion equations, Quart. appl. math., 40, 293-296, (1982) · Zbl 0506.35059 [13] Kaplan, S, On the growth of solutions of quasilinear parabolic equations, Comm. pure appl. math., 16, 305-330, (1963) · Zbl 0156.33503 [14] Ladyzenskaja, O.A; Solonnikov, V.A; Ural’cera, N.N, Linear and quasilinear equations of parabolic type, (), Translations of [15] {\scM. Langlais and D. Phillips}, to appear. [16] {\scP.-L. Lions}, “Asymptotic Behavior of Some Nonlinear Heat Equations,” T.S.R. 2134, Mathematics Research Center, Madison, Wisconsin. [17] Matano, H, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS Kyoto univ., 15, 401-454, (1979) · Zbl 0445.35063 [18] Ni, W.-M, Uniqueness of solutions of nonlinear Dirichlet problems, J. differential equations, 50, 289-304, (1983) · Zbl 0476.35033 [19] {\scW.-M. Ni and R. Nussbaum}, Uniqueness and nonuniqueness for positive radial solutions of Δu + f;(u, r) = 0, Comm. Pure Appl. Math., in press. [20] Pohozaev, S.I, Eigenfunctions of the equation δu + λf(u) = 0, Soviet math. dokl., 5, 1408-1411, (1965) · Zbl 0141.30202 [21] Rothe, F, Uniform bounds from bounded Lp functionals in reaction-diffusion equations, J. differential equations, 45, 207-233, (1982) · Zbl 0457.35043 [22] Sacks, P.E, Continuity of solutions of a singular parabolic equation, Nonlinear anal., 7, 387-409, (1983) · Zbl 0511.35052 [23] {\scP. E. Sacks}, Global behavior for a class of nonlinear evolution equations, to appear. · Zbl 0572.35062 [24] Weissler, F, Local existence and non-existence for semilinear parabolic equations in Lp, Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.