×

zbMATH — the first resource for mathematics

Maximal space regularity for abstract linear non-autonomous parabolic equations. (English) Zbl 0563.47028
Let E be a Banach space and \(\{\) A(t)\(\}\), \(t\in [0,T]\) a family of closed linear operators in E. The authors study the linear non-autonomous Cauchy problem \[ u'(t)-A(t)u(t)=f(t)\quad for\quad t\in [0,T],\quad u(0)=x,\quad x\in E,\quad f\in C([0,T],E); \] A(t) is the infinitesimal operator of an analytic semigroup, not necessarily strongly continuous at 0, with domains not depending on t. The abstract regularity is studied by means of interpolation spaces.
Reviewer: J.de Graaf

MSC:
47D03 Groups and semigroups of linear operators
46M35 Abstract interpolation of topological vector spaces
35K55 Nonlinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Acquistapace, P.; Terreni, B., Some existence and regularity results for abstract non-autonomous parabolic equations, J. math. anal. appl., 99, 9-64, (1984) · Zbl 0555.34051
[2] Acquistapace, P.; Terreni, B., On the abstract non-autonomous parabolic Cauchy problem in the case of constant domains, (), in press · Zbl 0952.49029
[3] Butzer, P.L.; Berens, H., Semi-groups of operators and approximation, (1969), Springer-Verlag Berlin
[4] Da Prato, G.; Grisvard, P., Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. math. pures appl., 54, 303-387, (1975) · Zbl 0315.47009
[5] Da Prato, G.; Grisvard, P., Équations d’évolution abstraites non linéaires de type parabolyque, Ann. mat. pura appl., 120, 329-396, (1979), (4) · Zbl 0471.35036
[6] Kato, T., Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, (), 467-468 · Zbl 0095.10502
[7] Lions, J.L., Théorèmes de trace et d’interpolation, I, Ann. scuola norm. sup. Pisa (III), 13, 389-403, (1959) · Zbl 0097.09502
[8] Lions, J.L.; Peetre, J., Sur une classe d’espaces d’interpolation, Inst. hautes études sci. publ. math., 19, 5-68, (1964) · Zbl 0148.11403
[9] Lunardi, A., Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, (), in press · Zbl 0568.47035
[10] Martin, R.H., Nonlinear operators and differential equations in Banach spaces, (1976), Wiley New York
[11] Pazy, A., Semi-groups of linear operators and applications to partial differential equations, () · Zbl 0516.47023
[12] \scE. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., in press. · Zbl 0589.47042
[13] Sobolevskii, P.E.; Sobolevskii, P.E., On equations of parabolic type in Banach space, (), Amer. math. soc. transl., 49, 1-62, (1965), English transl. · Zbl 0278.34054
[14] Tanabe, H., On the equations of evolution in a Banach space, Osaka math. J., 12, 145-166, (1960) · Zbl 0098.31202
[15] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · 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.