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The third boundary-value problem for parabolic differential-difference equations. (English. Russian original) Zbl 1176.35177
J. Math. Sci., New York 153, No. 5, 591-611 (2008); translation from Sovrem. Mat., Fundam. Napravl. 21, 114-132 (2007).
This paper is devoted to study the solvability and smoothness of strong solutions of the third boundary-value problem for parabolic differential-difference equations with translations with respect to spatial variables. The methods applied are based on the theory of elliptic functional-differential equations.

MSC:
35R10 Functional partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
47D06 One-parameter semigroups and linear evolution equations
47N20 Applications of operator theory to differential and integral equations
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References:
[1] A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basel-Boston-Berlin, (1994).
[2] G. A. Kamenskii and A. L. Skubachevskii, Linear Boundary Value Problems for Differential-Difference Equations [in Russian], MAI, Moscow (1990).
[3] T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972). · Zbl 0247.47009
[4] J.-L. Lions, Optimal Control of Systems That Are Governed by Partial Differential Equations [Russian translation], Mir, Moscow (1972).
[5] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications [Russian translation], Mir, Moscow (1971). · Zbl 0212.43801
[6] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. 2, Springer-Verlag, New York-Heidelberg-Berlin (1972). · Zbl 0223.35039
[7] V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1983).
[8] A. V. Razgulin, ”Rotational multi-petal waves in optical systems with 2-D feedback,” Chaos in Optics. Proceedings SPIE, 2039, 342–352 (1993).
[9] A. M. Selitskii, ”The third boundary value problem for parabolic differential-difference equations in the one-dimensional case,” Funct. Differ. Equ., To appear. · Zbl 1135.35089
[10] A. M. Selitskii and A. L. Skubachevskii, ”The second boundary-value problem for parabolic differential-difference equations,” Tr. Semin. im. I. G. Petrovskogo, To appear. · Zbl 1173.35714
[11] R. V. Shamin, ”Spaces of initial data for differential equations in a Hilbert space,” Sb. Math., 194, No. 9–10, 1411–1426 (2003). · Zbl 1073.34070 · doi:10.1070/SM2003v194n09ABEH000770
[12] R. V. Shamin, ”Nonlocal parabolic problems with the support of nonlocal terms inside a domain,” Funct. Differ. Equ., 10, No. 1–2, 307–314 (2003). · Zbl 1050.35124
[13] A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin, (1997). · Zbl 0946.35113
[14] A. L. Skubachevskii, ”On the Hopf bifurcation for a quasilinear parabolic functional-differential equation,” Differ. Equ., 34, No. 10, 1395–1402 (1998). · Zbl 0963.35018
[15] A. L. Skubachevskii, ”Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics,” Nonlinear Anal, 32, No. 2, 261–278 (1998). · Zbl 0916.35127 · doi:10.1016/S0362-546X(97)00476-8
[16] A. L. Skubachevskii and R. V. Shamin, ”The first mixed problem for a parabolic differential-difference equation,” Math. Notes, 66, No. 1–2, 113–119 (1999). · Zbl 0944.35106 · doi:10.1007/BF02674077
[17] A. L. Skubachevskii and R. V. Shamin, ”The mixed boundary value problem for parabolic differential-difference equations,” Funct. Differ. Equ., 8, No. 3–4, 407–424 (2001). · Zbl 1054.35119
[18] A. L. Skubachevskii and E. L. Tsvetkov, ”The second boundary value problem for elliptic differential-difference equations,” Differ. Equ., 25, No. 10, 1245–1254 (1989). · Zbl 0703.35051
[19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Ambrosius Barth, Heidelberg (1995). · Zbl 0830.46028
[20] V. V. Vlasov, ”On the solvability and properties of solutions of functional-differential equations in a Hilbert space,” Sb. Math., 186, No. 8, 1147–1172 (1995). · Zbl 0951.34052 · doi:10.1070/SM1995v186n08ABEH000060
[21] V. V. Vlasov, ”On the solvability and estimates for the solutions of functional-differential equations in Sobolev spaces,” Proc. Steklov Inst. Math., 4(227), 104–115 (1999). · Zbl 0985.35100
[22] M. A. Vorontsov, N. G. Iroshnikov, and R. L. Abernathy, ”Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation,” Chaos Solitons Fractals, 4, No. 8–9, 1701–1716 (1994). · Zbl 0813.35128 · doi:10.1016/0960-0779(94)90105-8
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