Nonautonomous parabolic equations involving measures.(English)Zbl 1159.35386

J. Math. Sci., New York 130, No. 4, 4780-4802 (2005); and Zap. Nauchn. Semin. POMI 306, 16-52 (2003).
Summary: In the first part of this paper, we study abstract parabolic evolution equations involving Banach space-valued measures. These results are applied in the second part to second-order parabolic systems under minimal regularity hypotheses on the coefficients.

MSC:

 35K90 Abstract parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K40 Second-order parabolic systems 35R05 PDEs with low regular coefficients and/or low regular data 47D06 One-parameter semigroups and linear evolution equations
Full Text:

References:

 [1] H. Amann, ”Existence and regularity for semilinear parabolic evolution equations,” Ann. Scuola Norm. Sup. Pisa, Ser. IV, 11, 593–676 (1984). · Zbl 0625.35045 [2] H. Amann, ”Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,” Diff. Int. Equat., 3, No.1, 13–75 (1990). · Zbl 0729.35062 [3] H. Amann, ”Nonhomogeneous linear and quasilinear elliptic and parabolic boundary-value problems,” in: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Vol. 133 of Teubner-Texte Math., Teubner, Stuttgart (1993), pp. 9–126. [4] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Abstract Linear Theory, Monographs in Mathematics, 89, Birkhauser (1995). · Zbl 0819.35001 [5] H. Amann, ”Linear parabolic problems involving measures,” Rev. R. Acad. Cien. Serie A. Mat. (RACSAM), 95, 85–119 (2001). · Zbl 1018.35049 [6] H. Amann, ”Maximal regularity for nonautonomous evolution equations,” Preprint (2003). · Zbl 1072.35103 [7] H. Amann, ”Maximum principles and principal eigenvalues,” (2003) (to appear). · Zbl 1090.35090 [8] H. Amann, M. Hieber, and G. Simonett, ”Bounded H calculus for elliptic operators,” Diff. Int. Equat., 7, 613–653 (1994). · Zbl 0799.35060 [9] H. Amann and P. Quittner, ”Control problems governed by semilinear parabolic equations with nonmonotone nonlinearities and low regularity data,” Preprint (2003). · Zbl 1106.49005 [10] H. Amann and P. Quittner, ”Semilinear parabolic equations involving measures and low-regularity data,” Trans. Amer. Math. Soc., 356, 1045–1119 (2004). · Zbl 1072.35094 [11] J. Bergh and J. Lofstrom, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin (1976). [12] R. Denk, G. Dore, M. Hieber, J. Pruss, and A. Venni, ”New thoughts on old results of R. T. Seeley,” Preprint (2003). [13] R. Denk, M. Hieber, and J. Pruss, ” $$\mathcal{R}$$ -boundedness, Fourier multipliers, and problems of elliptic and parabolic type,” Mem. Amer. Math. Soc., (2003) (to appear). · Zbl 1274.35002 [14] G. Dore, ”L p regularity for abstract differential equations,” Lect. Notes Math., 1540, 25–38 (1993). · Zbl 0818.47044 [15] P. E. Sobolevskii, ”Coerciveness inequalities for abstract parabolic equations,” Soviet Math. Dokl., 5, 894–897 (1964). · Zbl 0149.36001 [16] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam (1978). · Zbl 0387.46032
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.