Almost periodic (type) solutions to parabolic Cauchy inverse problems. (English) Zbl 1246.35209

Yang, Fenglin; Zhang, Chuanyi

Abstr. Appl. Anal. 2012, Article ID 180894, 17 p. (2012).

Summary: We first show the existence and uniqueness of (pseudo) almost periodic solutions of some types of parabolic equations. Then, we apply the results to a type of Cauchy parabolic inverse problems and show the existence, uniqueness, and stability.

MSC:

35R30	Inverse problems for PDEs
35K10	Second-order parabolic equations
35B10	Periodic solutions to PDEs

References:

[1]	C. Y. Zhang, “Pseudo-almost-periodic solutions of some differential equations,” Journal of Mathematical Analysis and Applications, vol. 181, no. 1, pp. 62-76, 1994. · Zbl 0796.34029
[2]	C. Y. Zhang, “Integration of vector-valued pseudo-almost periodic functions,” Proceedings of the American Mathematical Society, vol. 121, no. 1, pp. 167-174, 1994. · Zbl 0818.42003
[3]	R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532-3554, 2010. · Zbl 1248.34004
[4]	E. Ait Dads and O. Arino, “Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations,” Nonlinear Analysis A, vol. 27, no. 4, pp. 369-386, 1996. · Zbl 0855.34055
[5]	E. Ait Dads and K. Ezzibi, “Positive pseudo almost periodic solutions for some nonlinear delay integrable equation,” J. Cybemetics, vol. 6, pp. 134-145, 1994.
[6]	E. Ait Dads and K. Ezzinbi, “Pseudo almost periodic solutions of some delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 201, no. 3, pp. 840-850, 1996. · Zbl 0858.34055
[7]	E. A. Dads and K. Ezzinbi, “Existence of positive pseudo-almost-periodic solution for some nonlinear infinite delay integral equations arising in epidemic problems,” Nonlinear Analysis A, vol. 41, pp. 1-13, 2000. · Zbl 0964.45003
[8]	E. Ait Dads, K. Ezzinbi, and O. Arino, “Pseudo almost periodic solutions for some differential equations in a Banach space,” Nonlinear Analysis A, vol. 28, no. 7, pp. 1141-1155, 1997. · Zbl 0874.34041
[9]	A. I. Alonso, J. Hong, and R. Obaya, “Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences,” Applied Mathematics Letters, vol. 13, no. 2, pp. 131-137, 2000. · Zbl 0978.34039
[10]	A. I. Alonso, J. Hong, and R. Obaya, “Exponential dichotomy and trichotomy for difference equations,” Computers & Mathematics with Applications, vol. 38, no. 1, pp. 41-49, 1999. · Zbl 0939.39003
[11]	A. I. Alonso, J. Hong, and J. Rojo, “A class of ergodic solutions of differential equations with piecewise constant arguments,” Dynamic Systems and Applications, vol. 7, no. 4, pp. 561-574, 1998. · Zbl 0927.34028
[12]	B. Basit and C. Zhang, “New almost periodic type functions and solutions of differential equations,” Canadian Journal of Mathematics, vol. 48, no. 6, pp. 1138-1153, 1996. · Zbl 0880.43009
[13]	J. Hong and C. Núñez, “The almost periodic type difference equations,” Mathematical and Computer Modelling, vol. 28, no. 12, pp. 21-31, 1998. · Zbl 0992.39003
[14]	J. Hong and R. Obaya, “Ergodic type solutions of some differential equations,” in Differential Equations and Nonlinear Mechanics, vol. 528, pp. 135-152, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. · Zbl 0995.34048
[15]	J. Hong, R. Obaya, and A. S. Gil, “Exponential trichotomy and a class of ergodic solutions of differential equations with ergodic perturbations,” Applied Mathematics Letters, vol. 12, no. 1, pp. 7-13, 1999. · Zbl 0935.34044
[16]	J. Hong, R. Obaya, and A. M. Sanz, “Ergodic solutions via ergodic sequences,” Nonlinear Analysis A, vol. 40, pp. 265-277, 2000. · Zbl 0957.34054
[17]	J. Hong, R. Obaya, and A. Sanz, “Almost periodic type solutions of some differential equations with piecewise constant argument,” Nonlinear Analysis A, vol. 45, pp. 661-688, 2001. · Zbl 0996.34062
[18]	H.-X. Li, F.-I. Huang, and J.-Y. Li, “Composition of pseudo almost-periodic functions and semilinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 436-446, 2001. · Zbl 1047.47030
[19]	R. Yuan, “Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument,” Nonlinear Analysis A, vol. 41, pp. 871-890, 2000. · Zbl 1024.34068
[20]	C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, China, 2003. · Zbl 1068.34001
[21]	C. Y. Zhang, “Pseudo almost periodic solutions of some differential equations. II,” Journal of Mathematical Analysis and Applications, vol. 192, no. 2, pp. 543-561, 1995. · Zbl 0826.34040
[22]	C. Zhang, “Vector-valued pseudo almost periodic functions,” Czechoslovak Mathematical Journal, no. 3, pp. 385-394, 1997. · Zbl 0901.42005
[23]	C.-Y. Zhang, “Ergodicity and its applications. I. Basic properties,” Acta Analysis Functionalis Applicata, vol. 1, no. 1, pp. 28-39, 1999. · Zbl 0960.37002
[24]	C.-Y. Zhang, “Ergodicity and its applications. II. Averaging method of some dynamical systems,” Acta Analysis Functionalis Applicata, vol. 1, no. 2, pp. 146-159, 1999. · Zbl 0973.37005
[25]	C. Zhang, “Ergodicity and its applications in regularity and solutions of pseudo-almost periodic equations,” Nonlinear Analysis A, vol. 46, no. 4, pp. 511-523, 2001. · Zbl 1004.34033
[26]	C. Zhang and H. Yao, “Converse problems of Fourier expansion and their applications,” Nonlinear Analysis A, vol. 56, no. 5, pp. 761-779, 2004. · Zbl 1044.42006
[27]	J. Bourgain, “Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations,” Geometric and Functional Analysis, vol. 6, no. 2, pp. 201-230, 1996. · Zbl 0872.35007
[28]	C. Corduneanu, Almost Periodic Functions, Interscience Publishers, Chelsea, NY, USA, 1st edition, 1968. · Zbl 0175.09101
[29]	C. Corduneanu, Almost Periodic Functions, Interscience Publishers, Chelsea, NY, USA, 2nd edition, 1989. · Zbl 0672.42008
[30]	A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974. · Zbl 0325.34039
[31]	B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, UK, 1982. · Zbl 0499.43005
[32]	W. Shen, “Traveling waves in time almost periodic structures governed by bistable nonlinearties, I stability and uniqueness; II. existence,” Journal of Differential Equations, vol. 159, no. 1, pp. 1-110, 1999. · Zbl 0939.35016
[33]	F. Yang and C. Zhang, “Slowly oscillating solutions of parabolic inverse problems,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1238-1258, 2007. · Zbl 1127.35085
[34]	F. Yang and C. Zhang, “Slowly oscillating solutions of a parabolic inverse problem: boundary value problems,” Boundary Value Problems, vol. 2010, Article ID 471491, 12 pages, 2010. · Zbl 1207.35283
[35]	C. Zhang and F. Yang, “Remotely almost periodic solutions of parabolic inverse problems,” Nonlinear Analysis A, vol. 65, no. 8, pp. 1613-1623, 2006. · Zbl 1106.35142
[36]	C. Zhang and F. Yang, “Pseudo almost periodic solutions to parabolic boundary value inverse problems,” Science in China A, vol. 51, no. 7, pp. 1203-1214, 2008. · Zbl 1166.35303
[37]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. · Zbl 0144.34903
[38]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[39]	B. Guo, Inverse Problem of Parabolic Partial Dfferential Equations, Science and Technology Press, Harbin, China, 1988.
[40]	H. W. Engl, M. Hanke, and A. Neubauer, Rugularrization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. · Zbl 0859.65054