Geodesic
On coercive solvability of parabolic equations with variable operator
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 61 (2016), pp. 164-181.

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In a Banach space $E$, the Cauchy problem $$ v'(t)+A(t)v(t)=f(t)\quad (0\leq t\leq1),\qquad v(0)=v_0 $$ is considered for a differential equation with linear strongly positive operator $A(t)$ such that its domain $D=D(A(t))$ is everywhere dense in $E$ independently off $t$ and $A(t)$ generates an analytic semigroup $\exp\{-sA(t)\}$ ($s\geq0$). Under some natural assumptions on $A(t)$, we establish coercive solvability of the Cauchy problem in the Banach space $C_0^{\beta,\gamma}(E)$. We prove a stronger estimate of the solution compared to estimates known earlier, using weaker restrictions on $f(t)$ and $v_0$. 
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A. R. Hanalyev. On coercive solvability of parabolic equations with variable operator. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 61 (2016), pp. 164-181. http://geodesic.mathdoc.fr/item/CMFD_2016_61_a5/

[1] A. Ashyralyev, A. Khanalyev, “Coercive estimate in Hölder norms for parabolic equations with variable operator”, Modelling of mining processes for gas deposits and applied problems of theoretical gas-hydrodynamics, Ylym, Ashgabat, 1998, 154–162 (in Russian)

[2] M. A. Krasnosel'skiy, P. P. Zabreyko, E. I. Pustyl'nik, and P. E. Sobolevskiy, Integral Operators in Spaces of Summable Functions, Nauka, Moscow, 1966 (in Russian) | MR

[3] S. G. Kreyn, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967 (in Russian) | MR

[4] S. G. Kreyn, M. I. Khazan, “Differential equations in Banach space”, Totals Sci. Tech. Ser. Math. Anal., 21, 1983, 130–264 (in Russian) | MR | Zbl

[5] P. E. Sobolevskiy, “On equations of parabolic type in a Banach space”, Proc. Moscow Math. Soc., 10, 1961, 297–350 (in Russian) | MR | Zbl

[6] P. E. Sobolevskiy, “Coercivity inequalities for abstract parabolic equations”, Rep. Acad. Sci. USSR, 157:1 (1964), 52–55 (in Russian) | MR

[7] P. E. Sobolevskiy, “On fractional norms generated by an unbounded operator in Banach space”, Progr. Math. Sci., 19:6 (1964), 219–222 (in Russian)

[8] V. A. Rudetskiy, Dep. VINITI No. 34-85, VGU, 1984, Rzhmat 751102, 1985 (in Russian)

[9] Ashyralyev A., Hanalyev A., Sobolevskii P. E., “Coercive solvability of the nonlocal boundary-value problem for parabolic differential equations”, Abstr. Appl. Anal., 6:1 (2001), 53–61 | DOI | MR | Zbl

[10] Ashyralyev A., Sobolevskii P. E., New difference schemes for partial differential equations, Birkhäuser, Basel–Boston–Berlin, 2004 | MR | Zbl