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The semiflow of a reaction diffusion equation with a singular potential. (English) Zbl 1178.35221

Let Ω be an open set in N and assume that λ,μ,γ are positive parameters. This paper deals with the study of dynamics of the nonlinear parabolic equation

t φ-Δφ-μ |x| 2 φ=λφ-|φ| 2γ φ,xΩ,t>0,

subject to the initial condition

φ(x,0)=φ 0 (x),xΩ

and the boundary condition


The authors establish bifurcation results and they study the asymptotic behaviour of solutions as μμ * , where μ * denotes the optimal constant for the Hardy-Poincaré inequality. The proofs combine variational methods, elliptic and parabolic estimates and related inequalities involving singular terms.

35K67Singular parabolic equations
35B40Asymptotic behavior of solutions of PDE
26D10Inequalities involving derivatives, differential and integral operators
35B41Attractors (PDE)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K58Semilinear parabolic equations
35K20Second order parabolic equations, initial boundary value problems
35B32Bifurcation (PDE)
[1]Ball J.M.: On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations. J. Differ. Equ. 27, 224–265 (1978) · Zbl 0376.35002 · doi:10.1016/0022-0396(78)90032-3
[2]Ball J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004) · Zbl 1056.37084 · doi:10.3934/dcds.2004.10.31
[3]Baras P., Goldstein J.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284, 121–139 (1984) · doi:10.1090/S0002-9947-1984-0742415-3
[4]Brezis H., Cabre X.: Some simple nonlinear PDE’s without solutions. Boll. Unione Mat. Ital. Sez. B 1, 223–262 (1998)
[5]Brezis H., Dupaigne L., Tesei A.: On a semilinear elliptic equation with inverse-square potential. Sel. Math. New Ser. 11, 1–7 (2005) · Zbl 1161.35383 · doi:10.1007/s00029-005-0003-z
[6]Brezis H., Vázquez J.L.: Blowup solutions of some nonlinear elliptic problems. Rev. Math. Univ. Complutense Madrid 10, 443–469 (1997)
[7]Brown, K.J.: Local and global bifurcation results for a semilinear boundary value problem. J. Differ. Equ. (2007). doi: 1016/j.jde.2007.05.013
[8]Brown K.J., Stavrakakis N.M.: Global bifurcation results for a semilinear elliptic equation on all of N . Duke Math. J. 85, 77–94 (1996) · Zbl 0862.35010 · doi:10.1215/S0012-7094-96-08503-8
[9]Busca J., Jendoubi M.A., Pol áčik P.: Convergence to equilibrium for semilinear parabolic problems in N . Comm. Partial Differ. Equ. 27, 1793–1814 (2002) · Zbl 1021.35013 · doi:10.1081/PDE-120016128
[10]Cabré, X., Martel, Y.: Existence versus explosion instantané pour des equations de la chaleur linéaires avec potentiel singulier. C.R. Acad. Sci. Paris 329, 973–978 (1999)
[11]Cazenave, T., Haraux, A.: Introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications 13 (1998)
[12]Chaves M., Azorero J.G.: On bifurcation and uniqueness results for some semilinear elliptic equations involving a singular potential. J. Eur. Math. Soc. (JEMS) 8(2), 229–242 (2006) · Zbl 05053361 · doi:10.4171/JEMS/49
[13]Dávila J., Dupaigne L.: Comparison principles for PDEs with a singular potential. Proc. R. Soc. Edinburgh 133, 61–83 (2003) · Zbl 1040.35006 · doi:10.1017/S0308210500002286
[14]Esteban M.J., Giacomoni J.: Existence of global branches of positive solutions for semilinear elliptic degenerate problems. J. Math. Pures Appl. 79, 715–740 (2000) · Zbl 0952.35060 · doi:10.1016/S0021-7824(00)00104-5
[15]Filippas S., Tertikas A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002) · Zbl 1030.26018 · doi:10.1006/jfan.2001.3900
[16]Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 Edition, Springer, Berlin-Heidelberg-New York (2001)
[17]Hale, J.K.: Asymptotic behaviour of dissipative systems. Math. Surv. Monogr. 25 Amer. Math. Soc., Providence, R.I. (1988)
[18]Karachalios N.I.: Weyl’s type estimates on the eigenvalues of critical Schrödinger operators. Lett. Math. Phys. 83 (2), 189–199 (2008) · Zbl 1162.35057 · doi:10.1007/s11005-007-0218-3
[19]Karachalios N.I., Zographopoulos N.B.: On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence. Calc. Var. Partial Differ. Equ. 25(3), 361–393 (2006) · Zbl 1090.35035 · doi:10.1007/s00526-005-0347-4
[20]Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44, Springer New York (1983)
[21]Pucci P., Serrin J.: The strong maximum principle revisited (review). J. Differ. Equ. 196, 1–66 (2004) · Zbl 1109.35022 · doi:10.1016/j.jde.2003.05.001
[22]Rabinowitz P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[23]Tertikas A., Zographopoulos N.B.: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math. 209(2), 407–459 (2007) · Zbl 1160.26010 · doi:10.1016/j.aim.2006.05.011
[24]Temam R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)
[25]Vazquez J.L., Zuazua E.: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173, 103–153 (2000) · Zbl 0953.35053 · doi:10.1006/jfan.1999.3556
[26]Zeidler, E.: Nonlinear functional analysis and its applications vols. I, II, (Fixed Point Theorems, Monotone Operators). Springer, Berlin (1990)