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The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. (English) Zbl 0953.35053
The paper is concerned with the well-posedness and the asymptotic behavior of solutions of the heat equation with the potential: $$u_t=\Delta u +\lambda |x|^{-2} u$$ for the Cauchy-Dirichlet problem in a bounded domain as well as for the Cauchy problem in $${\mathbb{R}}^N$$. In the case of the bounded domain, the use of an improved form of the so-called Hardy-Poincaré inequality allows the authors to prove the exponential stabilisation towards a solution in separated variables. In $${\mathbb{R}}^N$$, they first establish a new version of the Hardy-Poincaré inequality, and next, they show the stabilisation towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This work complements and explains the paper by P. Baras and J. A. Goldstein [Trans. Am. Math. Soc. 284, 121-139 (1984; Zbl 0556.35063)] on the existence of global solutions and blow-up for this equation. In the present article, the sign restriction on the data and solutions is removed, the functional framework for the well-posedness is described, and the asymptotic rates are calculated. Examples of non-uniqueness are also given.

##### MSC:
 35K10 Second-order parabolic equations 35K65 Degenerate parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
Zbl 0556.35063
Full Text:
##### References:
 [1] Baras, P.; Goldstein, J., The heat equation with a singular potential, Trans. amer. math. soc., 284, 121-139, (1984) · Zbl 0556.35063 [2] Bebernes, J.; Eberly, D., Mathematical problems from combustion theory, Math. sci., 83, (1989), Springer-Verlag New York · Zbl 0692.35001 [3] Bénilan, Ph.; Boccardo, L.; Gallouet, Th.; Gariepy, R.; Pierre, M.; Vazquez, J.L., An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. scuola norm. sup. Pisa, 22, 241-273, (1995) · Zbl 0866.35037 [4] Bénilan, Ph.; Bouhsiss, F., Une remarque sur l’unicité des solutions pour l’opérateur de Serrin, C. R. acac. sci. Paris I, 325, 611-616, (1997) · Zbl 0892.35049 [5] Bénilan, Ph.; Brezis, H.; Crandall, M.G., A semilinear equation in L1($$R$$N), Ann. scuola norm. sup. Pisa cl. sci. (4), 2, 523-555, (1975) · Zbl 0314.35077 [6] Berger, M.; Gauduchon, P.; Mazet, E., Le spèctre d’une variété riemannienne, (1971), Springer-Verlag Berlin [7] Brezis, H.; Marcus, M., Hardy’s inequality revisited, Ann. scuola norm. sup. Pisa cl. sci., 25, 217-237, (1997) · Zbl 1011.46027 [8] Brezis, H.; Vazquez, J.L., Blow-up solutions of some nonlinear elliptic equations, Rev. mat. complut., 10, 443-469, (1997) · Zbl 0894.35038 [9] Cabré, X.; Martel, Y., Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. acad. sci. Paris, 329, 973-978, (1999) · Zbl 0940.35105 [10] Davies, E.B., Heat kernels and spectral theory, Cambridge tracts in mathematics, 92, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006 [11] Dold, J.W.; Galaktionov, V.A.; Lacey, A.A.; Vazquez, J.L., Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. scuola norm. sup. Pisa cl. sci. (4), 26, 663-687, (1998) · Zbl 0920.35080 [12] Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear anal., 11, 1103-1133, (1987) · Zbl 0639.35038 [13] Escobedo, M.; Zuazua, E., Large time behaviour for convection-diffusion equations in $$R$$^{N}, J. funct. anal., 100, 119-161, (1991) · Zbl 0762.35011 [14] Galaktionov, V.; Vazquez, J.L., Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. pure appl. math., 1, 1-67, (1997) · Zbl 0874.35057 [15] Gilbarg, D.; Trudinger, N.S., Elliptic differential equations of the second order, (1983), Springer-Verlag New York/Berlin · Zbl 0691.35001 [16] Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques et applications, 13, (1993), Springer-Verlag New York/Berlin · Zbl 0797.58005 [17] Lions, J.L.; Magnenes, E., Problèmes aux limites non homogènes, (1968), Dunod Paris [18] Maz’ja, V.G., Sobolev spaces, (1985), Springer-Verlag New York/Berlin [19] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1992), Springer-Verlag New York · Zbl 0516.47023 [20] Peral, I.; Vazquez, J.L., On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. rational mech. anal., 129, 201-224, (1995) · Zbl 0821.35080 [21] Prignet, A., Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures, Rend. mat. appl., 7, 189-206, (1995) · Zbl 0843.35127 [22] Reed, M.; Simon, B., Methods of modern mathematical physics, (1979), Academic Press New York [23] Serrin, J., Pathological solution of elliptic differential equation, Ann. scuola norm. sup. Pisa, 17, 385-387, (1964) · Zbl 0142.37601 [24] Vazquez, J.L., Domain of existence and blowup for the exponential reaction-diffusion equation, Indiana univ. math. J., 48, 677-709, (1999) · Zbl 0928.35080 [25] Vazquez, J.L.; Yarur, C., Isolated singularities of the solutions of the Schrödinger equation with a radial potential, Arch. rational mech. anal., 98, 251-284, (1987) · Zbl 0645.35017 [26] Véron, L., Coercivité et propriétés régularisantes des semi-groupes non-linéaires dans LES espaces de Banach, Ann. fac. sci. Toulouse, 1, 171-200, (1979) [27] Zeidler, E., Applied functional analysis, Appl. math. sci., 108, (1995), Springer-Verlag New York
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