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On asymptotics of solutions of parabolic equations with nonlocal high-order terms. (English. Russian original) Zbl 1115.35140
J. Math. Sci., New York 135, No. 1, 2695-2720 (2006); translation from Tr. Semin. Im. I. G. Petrovskogo 25, 143-183 (2005).
Summary: We study the Cauchy problem for second-order differential-difference parabolic equations $\frac{\partial u}{\partial t}= \frac{\partial^2u}{\partial x^2}+ \sum_{k=1}^m a_k \frac{\partial^2u}{\partial x^2} (x+h_k,t) \quad\text{in }\mathbb R^1\times(0,\infty)$ containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-time behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained.

##### MSC:
 35R10 Partial functional-differential equations 35B40 Asymptotic behavior of solutions to PDEs 35C15 Integral representations of solutions to PDEs 35A08 Fundamental solutions to PDEs 35B35 Stability in context of PDEs
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##### References:
 [1] R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press, New York (1963). · Zbl 0105.06402 [2] J. Hale, Theory of Functional Differential Equations, Springer, New York (1984). · Zbl 1092.34500 [3] A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhauser, Basel (1997). · Zbl 0946.35113 [4] K. Kunisch and W. Schappacher, ”Necessary conditions for partial differential equations with delay to generate C 0-semigroups,” J. Differential Equations, 50, No. 1, 49–79 (1983). · Zbl 0533.35082 [5] W. Desch and W. Schappacher, ”Spectral properties of finite-dimensional perturbed linear semigroups,” J. Differential Equations, 59, No. 1, 80–102 (1985). · Zbl 0575.34049 [6] V. V. Vlasov, ”On a class of differential-difference equations in a Hilbert space and some spectral questions,” Russian Acad. Sci. Dokl. Math., 46, No. 3, 458–462 (1993). [7] V. M. Borok and E. S. Viglin, ”The uniqueness of the solution of the fundamental initial problem for partial differential equations with a deviating argument,” Differential Equations, 13, No. 7, 848–854 (1977). · Zbl 0409.35075 [8] A. I. Daševskii, ”A boundedness criterion for the solutions of linear difference-differential equations with retarded argument in Banach spaces,” Differential Equations, 13, No. 8, 1054–1056 (1977). [9] A. Inone, T. Miyakawa, and K. Yoshida, ”Some properties of solutions for semilinear heat equations with time lag,” J. Differential Equations, 24, No. 3, 383–396 (1977). · Zbl 0345.35057 [10] G. Di Blasio, K. Kunisch, and E. Sinestrari, ”L 2-regularity for parabolic partial integrodifferential equations with delay in highest-order derivatives,” J. Math. Anal. Appl., 102, No. 1, 38–57 (1984). · Zbl 0538.45007 [11] V. V. Vlasov and V. Zh. Sakbaev, ”The correct solvability of some differential-difference equations in the scale of Sobolev spaces,” Differential Equations, 37, No. 9, 1252–1260 (2001). · Zbl 1257.34058 [12] B. L. Gurevič, ”New types of fundamental and generalized function spaces and the Cauchy problem for systems of difference equations involving differential operations,” Dokl. Akad. Nauk SSSR, 108, No. 6, 1001–1003 (1956). · Zbl 0073.31501 [13] V. S. Rabinovich, ”The Cauchy problem for parabolic differential-difference operators with variable coefficients,” Differential Equations, 19, No. 6, 768–775 (1983). · Zbl 0548.35115 [14] A. L. Skubachevskii, ”On some properties of elliptic and parabolic differential-difference equations,” Russ. Math. Surv., 51, No. 1, 169–170 (1996). · Zbl 0876.35124 [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 [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 [17] A. B. Muravnik, ”On Cauchy problem for parabolic differential-difference equations,” Nonlinear Anal., 51, No. 2, 215–238 (2002). · Zbl 1010.35111 [18] A. B. Muravnik, ”On the Cauchy problem for differential-difference equations of the parabolic type,” Russian Acad. Sci. Dokl. Math., 66, No. 1, 107–110 (2002). · Zbl 1148.39300 [19] A. V. Razgulin, ”Rotational multi-petal waves in optical system with 2-D feedback,” Chaos in Optics. Proceedings SPIE, 2039, 342–352 (1993). [20] 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, and Fractals, 4, 1701–1716 (1994). · Zbl 0813.35128 [21] I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. 3. Theory of Differential Equations, Academic Press, New York (1967). · Zbl 0179.14501 [22] V. M. Borok and Ja. I. Zitomirskii, ”On the Cauchy problem for linear partial differential equations with linearly transformed argument,” Sov. Math. Dokl., 12, 1412–1416 (1971). · Zbl 0249.35003 [23] O. A. Ladyzhenskaya, ”On the uniqueness of the Cauchy problem solution for a linear parabolic equation,” Mat. Sb., 27 (59), No. 2, 175–184 (1950). [24] V. D. Repnikov and S. D. Ehjdel’man, ”Necessary and sufficient conditions for the establishment of a solution of the Cauchy problem,” Sov. Math. Dokl., 7, 388–391 (1966). · Zbl 0145.35701 [25] M. E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton (1981). [26] A. M. Il’in, A. S. Kalašnikov, and O. A. Oleinik, ”Second-order linear equations of parabolic type,” Russian Math. Surv., 17, No. 3, 1–146 (1962). · Zbl 0108.28401
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