Some boundary value problems for the Sobolev–type operator–differential equations
We consider the solvability of boundary value problems for operator-differential equation of the form $Bu_t-Lu=f,$ where $X$ is a Banach space, $B,\,L\,:\,X\rightarrow X$ are closed operators such that $D(L)\subset D(B)$ ($D(L), D(B)$ are domains of the corresponding operators), with boundary conditions $Bu(0) = Bu_0$ or $\int\limits^T_0 Bu(\tau)d\sigma(\tau)=Bu_0,$ where $\sigma$ is a function of bounded variation. Some well-known results on solvability of initial boundary value problems for operator-differential equations of Sobolev type are refined in the case of arbitrary decrease (growth) of the resolvent of the corresponding linear pencil. Existence and uniqueness theorems of solutions to the Cauchy-type problems and general nonlocal boundary value problems are obtained and the maximal regularity of solutions is proven under certain conditions. The results rely on Mikhlin theorems for operator-valued Fourier multipliers. In contrast to the previous results, the function spaces are the Sobolev–Besov spaces.
 Favini A. and Yagi A., “Multivalued linear operators and degenerate evolution equations,” Ann. Mat. Pura Appl., IV. Ser., 163, 353–384 (1993).
 Favini A. and Yagi A., Degenerate Differential Equations in Banach Spaces, Marcel Dekker, Inc., New York (1999).
 Favini A. and Yagi A., “Quasilinear degenerate evolution equations in Banach spaces,” J. Evolution Equ., 4, 421–449 (2004).
 Bojovic D. R., Jovanovic B. S., and Matus P., “ On the strong stability of first-order operatordifferential equations,” Differ. Equ., 40, No. 5, 703–710 (2004).
 Melnikova I. V., “Cauchy problem for inclusion in Banach spaces and distribution spaces,” Sib. Adv. Math., 42, No. 4, 892–910 (2001).
 Melnikova I. V. and Al’shansky M. A., “Well-posedness of the Cauchy problem in a Banach space: regular and degenerate cases,” J. Math. Sci., 87, No. 4, 3732–3780 (1997).
 Abdulkerimli L. Sh. and Eminova Sh. L., “Well-posedness of a class of operator-differential equations,” Differ. Equ., 53, No. 10, 1288–1293 (2017).
 Sviridyuk G. A. and Fedorov V. E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht (2003).
 Pyatkov S. G. and Abasheeva N. L., “Solvability of boundary value problems for operatordifferential equations of mixed type,” Sib. Math. J., 41, No. 6, 1419–1435 (2000).
 Pyatkov S. G. and Abasheeva N. L., “Solvability of boundary value problems for operatordifferential equations of mixed type. Degenerate case,” Sib. Math. J., 43, No. 3, 678–693 (2002).
 Favini A., Sviridyuk G. A., and Manakova N. A., “Linear Sobolev type equations with relatively p-sectorial operators in space of "noises",” Abstr. Appl. Anal., 8 (2015).
 Fedorov V. E., Ivanova N. D., and Fedorova Yu. Yu., “A time-nonlocal problem for inhomogeneous evolution equations,” Sib. Math. J., 55, No. 4, 882–897 (2014).
 Fedorov V. E., “Degenerate strongly continuous semigroups of operators [in Russian],” Algebra Anal., 12, No. 3, 173–200 (2000).
 Fedorov V. E. and Sagadeeva M. A., “Existence of exponential dichotomies of some classes of degenerate linear equations [in Russian],” Comput. Technol., 11, No. 2, 82–92 (2006).
 Tikhonov I. V., “Uniqueness theorems in linear non-local problems for abstract differential equations,” Izv. RAN, Ser. Math., 67, No. 2, 133–166 (2003).
 Denk R., Hieber M., and Pruss J., “R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type,” Mem. Amer. Math. Soc., 166, No. 788 (2003).
 Denk R. and Krainer T., “R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators,” Manuscr. Math., 124, No. 3, 319–342 (2007).
 Kunstman C. and Weis L., “Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and H∞-functional calculus,” Lect. Notes Math., 1855, 65–311 (2004).
 Denk R., Hieber M., and Pruss J., “Optimal $L_p$ - $L_q$-estimates for parabolic boundary value problems with inhomogeneous data,” Math. Z., 257, No. 1, 93–224 (2007).
 Grisvard P., “Commutative de deux functeurs d’interpolation et applications,” J. Math. Pures Appl., 45, No. 2. 143–206 (1966).
 Grisvard P., “Equations differentielles abstraites,” Ann. Sci. Ec. Norm. Super., IV Ser., 2, 311–395 (1969).
 Triebel H., Interpolation Theory, Function spaces, Diferential operators [in Russian], Mir, Moscow (1980).
 Da Prato G. and Grisvard P., “Commutative de deux functeurs d’interpolation et applications,” J. Math. Pures Appl., 54, No. 3, 305–387 (1975).
 Haase M., The Functional Calculus for Sectorial Operators. Birkhauser Verl., Basel; Boston; Berlin (2006) (Operator Theory: Adv. Appl.; 169).
This work is licensed under a Creative Commons Attribution 4.0 International License.