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On the Łojasiewicz–Simon gradient inequality. (English) Zbl 1036.26015
The author establishes in this paper a version of the Łojasiewicz-Simon inequality for functions which are not necessarily analytic. The abstract result is then applied to energy functionals of the form $$E(v)=\frac {1}{2}a(v,v)+\int_\Omega F(x,v)\,dx$$, defined on a Hilbert space which is compactly embedded into $$L^2(\Omega)$$. In the last part of the paper these results are applied to nonlinear evolution equations of the form $$u_t-\Delta u+| u| ^{p-1}u+\lambda u=0$$ in $${\mathbb R}\times\Omega$$. It is shown that in this example the underlying energy functional satisfies the Łojasiewicz-Simon inequality near the origin with Łojasiewicz exponent $$\theta =(p+1)^{-1}$$. The proofs are based on refined estimates and a careful analysis of several classes of differential operators.

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators 34D05 Asymptotic properties of solutions to ordinary differential equations 34G10 Linear differential equations in abstract spaces 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35L70 Second-order nonlinear hyperbolic equations
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##### References:
 [1] Aulbach, B., Continuous and discrete dynamics near manifolds of equilibria, Lecture notes in mathematics, Vol. 1058, (1984), Springer Berlin, New York, Heidelberg · Zbl 0535.34002 [2] Belaud, Y.; Helffer, B.; Véron, L., Long-time vanishing properties of solutions of some semilinear parabolic equations, Ann. inst. Henri Poincaré, anal. nonlinéaire, 18, 43-68, (2001) · Zbl 0983.35066 [3] Brunovský, P.; Polačik, P., On the local structure of ω-limit sets of maps, Z. angew. math. phys., 48, 976-986, (1997) · Zbl 0889.34048 [4] R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. Ser. A: Theory Methods, 2003, to appear. · Zbl 1033.34066 [5] Daners, D., Robin boundary value problems on arbitrary domains, Trans. amer. math. soc., 352, 4207-4236, (2000) · Zbl 0947.35072 [6] R. Dautray, J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. VI, INSTN: Collection Enseignement, Masson, Paris, 1988. [7] Feireisl, E.; Simondon, F., Convergence for semilinear degenerate parabolic equations in several space dimensions, J. dynam. differential equations, 12, 647-673, (2000) · Zbl 0977.35069 [8] Feireisl, E.; Takač, P., Long-time stabilization of solutions to the ginzburg – landau equations of superconductivity, Monatsh. math., 133, 197-221, (2001) · Zbl 0993.35045 [9] Hale, J.K.; Raugel, G., Convergence in gradient-like systems with applications to PDE, Z. angew. math. phys., 43, 63-124, (1992) · Zbl 0751.58033 [10] Haraux, A.; Jendoubi, M.A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. var. partial differential equations, 9, 95-124, (1999) · Zbl 0939.35122 [11] Haraux, A.; Jendoubi, M.A., Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptotic anal., 26, 21-36, (2001) · Zbl 0993.35017 [12] A. Haraux, M.A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, preprint, 2001. · Zbl 0993.35017 [13] A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evolution Equations, 2003, to appear. · Zbl 1036.35035 [14] Hoffmann, K.-H.; Rybka, P., Convergence of solutions to cahn – hillard equation, Commun. partial differential equations, 24, 1055-1077, (1999) · Zbl 0936.35032 [15] Huang, S.-Z.; Takač, P., Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear anal. ser. A: theory methods, 46, 675-698, (2001) · Zbl 1002.35022 [16] Jendoubi, M.A., Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. differential equations, 144, 302-312, (1998) · Zbl 0912.35028 [17] Jendoubi, M.A., A simple unified approach to some convergence theorems of L. Simon, J. funct. anal., 153, 187-202, (1998) · Zbl 0895.35012 [18] M.A. Jendoubi, P. Polačik, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Prépublications de l’Université de Versailles-St. Quentin, 74, 2001. [19] Kurdyka, K.; Parusiński, A., ωf-stratification of subanalytic functions and the łojasiewicz inequality, C. R. acad. sci. Paris, Sér. I, 318, 129-133, (1994) · Zbl 0799.32007 [20] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, pp. 87-89. [21] S. Łojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S., Bures-sur-Yvette, 1965. [22] S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, Seminari di Geometria, Bologna (1982/83), Universitá degli Studi di Bologna, Bologna, 1984, pp. 115-117. [23] Łojasiewicz, S.; Zurro, M.A., On the gradient inequality, Bull. Polish acad. sci. math., 47, 143-145, (1999) · Zbl 0932.32009 [24] Matano, H., Convergence of solutions of one-dimensional semilinear heat equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008 [25] P. Mironescu, V. Radulescu, Nonlinear Sturm-Liouville type problems with finite number of solutions, preprint, 1999. [26] Palis, J.; de Melo, W., Geometric theory of dynamical systems. an introduction, (1982), Springer New York, Heidelberg, Berlin [27] Polačik, P.; Rybakowski, K., Nonconvergent bounded trajectories in semilinear heat equations, J. differential equations, 124, 472-494, (1996) · Zbl 0845.35054 [28] P. Polačik, F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, preprint, 2001. [29] Reed, M.; Simon, B., Methods of modern mathematical physics, I: functional analysis, 2nd edition, (1980), Academic Press New York [30] Simon, L., Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. math., 118, 525-571, (1983) · Zbl 0549.35071 [31] Simon, L., Theorems on regularity and singularity of energy minimizing maps, lecture notes in mathematics ETH Zürich, (1996), Birkhäuser Basel [32] Taira, K., A mathematical analysis of thermal explosions, Int. J. math. mech. sci., 28, 581-607, (2001) · Zbl 1012.35034 [33] Taira, K.; Umezu, K., Stability in chemical reactor theory, (), 421-433 · Zbl 1004.35118 [34] Zeidler, E., Nonlinear functional analysis and its applications. I, (1990), Springer New York, Berlin, Heidelberg [35] Zelenyak, T.I., Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differential equations, 4, 17-22, (1968), (Transl. from Differ. Uravn. 4 (1968) 34-45) · Zbl 0232.35053
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