Monotonicity method applied to the complex Ginzburg-Landau and related equations. (English) Zbl 0995.35029

A new inequality in monotonicity methods is derived. It is based on the sectorial estimates of \(-\Delta\) in \(L^{p+1}\) and the nonlinear operator \(u\mapsto|u|^{p-1}u\) appearing in the equation.
By the inequality the global existence of unique strong solutions is established for the complex Ginzburg-Landau equation. The inequality also yields the global existence of unique strong solution for nonlinear Schrödinger type equation with monotone nonlinearity.
Reviewer: X.Xiang (Guiyang)


35K55 Nonlinear parabolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
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[1] Brezis, H., Analyse Fonctionnelle, Théorie et Applications (1983), Masson: Masson Paris · Zbl 0511.46001
[2] Brezis, H.; Crandall, M. G.; Pazy, A., Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. Pure Appl. Math., 23, 123-144 (1970) · Zbl 0182.47501
[3] Bu, C., On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B, 36, 313-324 (1994) · Zbl 0829.35119
[4] Crandall, M. G.; Liggett, T., Generation of semi-groups of nonlinear transformations in general Banach spaces, Amer. J. Math., 43, 265-298 (1971) · Zbl 0226.47038
[5] Doering, C. R.; Gibbon, J. D.; Levermore, C. D., Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71, 285-318 (1994) · Zbl 0810.35119
[6] Ginibre, J.; Velo, G., The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95, 191-228 (1996) · Zbl 0889.35045
[7] Ginibre, J.; Velo, G., The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys., 187, 45-79 (1997) · Zbl 0889.35046
[8] Hempel, R.; Voigt, J., On the \(L_p\)-spectrum of Schrödinger operators, J. Math. Anal. Appl., 121, 138-159 (1987) · Zbl 0622.35050
[9] Kato, T., Perturbation Theory for Linear Operators. Perturbation Theory for Linear Operators, Grundlehren math. Wissenschaften, 132 (1966), Springer-Verlag: Springer-Verlag Berlin/New York
[10] Kato, T., Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19, 508-520 (1967) · Zbl 0163.38303
[11] Kato, T., Accretive operators and nonlinear evolution equations in Banach spaces, Nonlinear Functional Analysis (1968), p. 138-161
[12] Levermore, C. D.; Oliver, M., The complex Ginzburg-Landau equation as a model problem, Dynamical Systems and Probabilistic Methods in Partial Differential Equations (1994), p. 141-190 · Zbl 0845.35003
[13] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[14] Liskevich, V. A.; Perelmuter, M. A., Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc., 123, 1097-1104 (1995) · Zbl 0826.47030
[15] Miyadera, I., Nonlinear Semigroups, Translations of Math. Mono. (1992), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0174.19402
[16] Okazawa, N., Singular perturbations of \(m\)-accretive operators, J. Math. Soc. Japan, 32, 19-44 (1980) · Zbl 0414.47023
[17] Okazawa, N., Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc., 113, 701-706 (1991) · Zbl 0752.35013
[18] Okazawa, N.; Yokota, T., Perturbation theory for \(m\)-accretive operators and generalized complex Ginzburg-Landau equations, J. Math. Soc. Japan, 54 (2002) · Zbl 1045.35080
[19] Pecher, H.; von Wahl, W., Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27, 125-157 (1979) · Zbl 0399.35030
[20] Shigeta, T., A characterization of \(m\)-accretivity and an application to nonlinear Schrödinger type equations, Nonlinear Anal. TMA, 10, 823-838 (1986) · Zbl 0613.47046
[21] Showalter, R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surv. Mono. (1997), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0870.35004
[22] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Math. Sci. (1988), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0662.35001
[23] Unai, A., Global \(C^1\)-solutions of time-dependent complex Ginzburg-Landau equations, Nonlinear Anal. TMA, 46, 329-334 (2001) · Zbl 0977.35064
[24] Unai, A.; Okazawa, N., Perturbations of nonlinear \(m\)-sectorial operators and time-dependent Ginzburg-Landau equations, Dynamical Systems and Differential Equations (1996), p. 259-266
[25] Yang, Y., On the Ginzburg-Landau wave equation, Bull. London Math. Soc., 22, 167-170 (1990)
[26] Zaag, H., Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. Henri Poincaré, Analyse non linéaire, 15, 581-622 (1998) · Zbl 0902.35050
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