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.

## References

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. 0796.34029 MR1257954 10.1006/jmaa.1994.1005 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. 0796.34029 MR1257954 10.1006/jmaa.1994.1005 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. 0818.42003 MR1186140 10.1090/S0002-9939-1994-1186140-8 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. 0818.42003 MR1186140 10.1090/S0002-9939-1994-1186140-8 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. MR2683811 10.1016/j.nonrwa.2010.01.002 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. MR2683811 10.1016/j.nonrwa.2010.01.002 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. 0855.34055 MR1393143 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. 0855.34055 MR1393143 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. 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. 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. 0858.34055 MR1400566 10.1006/jmaa.1996.0287 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. 0858.34055 MR1400566 10.1006/jmaa.1996.0287 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. 0964.45003 MR1759136 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. 0964.45003 MR1759136 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. 0874.34041 MR1422806 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. 0874.34041 MR1422806 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. 0978.34039 MR1751535 10.1016/S0893-9659(99)00176-7 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. 0978.34039 MR1751535 10.1016/S0893-9659(99)00176-7 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. 0939.39003 MR1697341 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. 0939.39003 MR1697341 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. 0927.34028 MR1664594 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. 0927.34028 MR1664594 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. 0880.43009 MR1426896 10.4153/CJM-1996-059-9 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. 0880.43009 MR1426896 10.4153/CJM-1996-059-9 J. Hong and C. Núñez, “The almost periodic type difference equations,”

*Mathematical and Computer Modelling*, vol. 28, no. 12, pp. 21–31, 1998. 0992.39003 MR1663845 10.1016/S0895-7177(98)00161-7 J. Hong and C. Núñez, “The almost periodic type difference equations,”*Mathematical and Computer Modelling*, vol. 28, no. 12, pp. 21–31, 1998. 0992.39003 MR1663845 10.1016/S0895-7177(98)00161-7 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. 0995.34048 MR1827589 10.1007/978-1-4613-0277-3_11 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. 0995.34048 MR1827589 10.1007/978-1-4613-0277-3_11 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. 0935.34044 MR1663409 10.1016/S0893-9659(98)00118-9 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. 0935.34044 MR1663409 10.1016/S0893-9659(98)00118-9 J. Hong, R. Obaya, and A. M. Sanz, “Ergodic solutions via ergodic sequences,”

*Nonlinear Analysis A*, vol. 40, pp. 265–277, 2000. 0957.34054 MR1768412 J. Hong, R. Obaya, and A. M. Sanz, “Ergodic solutions via ergodic sequences,”*Nonlinear Analysis A*, vol. 40, pp. 265–277, 2000. 0957.34054 MR1768412 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. 0996.34062 MR1841201 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. 0996.34062 MR1841201 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. 1047.47030 MR1815791 10.1006/jmaa.2000.7225 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. 1047.47030 MR1815791 10.1006/jmaa.2000.7225 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. 1024.34068 MR1764025 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. 1024.34068 MR1764025 C. Zhang,

*Almost Periodic Type Functions and Ergodicity*, Science Press, Beijing, China, 2003. MR2000981 1068.34001 C. Zhang,*Almost Periodic Type Functions and Ergodicity*, Science Press, Beijing, China, 2003. MR2000981 1068.34001 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. 0826.34040 MR1332227 10.1006/jmaa.1995.1189 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. 0826.34040 MR1332227 10.1006/jmaa.1995.1189 C. Zhang, “Vector-valued pseudo almost periodic functions,”

*Czechoslovak Mathematical Journal*, no. 3, pp. 385–394, 1997. 0901.42005 MR1461419 10.1023/A:1022492014464 C. Zhang, “Vector-valued pseudo almost periodic functions,”*Czechoslovak Mathematical Journal*, no. 3, pp. 385–394, 1997. 0901.42005 MR1461419 10.1023/A:1022492014464 C.-Y. Zhang, “Ergodicity and its applications. I. Basic properties,”

*Acta Analysis Functionalis Applicata*, vol. 1, no. 1, pp. 28–39, 1999. 0960.37002 MR1739236 C.-Y. Zhang, “Ergodicity and its applications. I. Basic properties,”*Acta Analysis Functionalis Applicata*, vol. 1, no. 1, pp. 28–39, 1999. 0960.37002 MR1739236 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. 0973.37005 MR1739253 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. 0973.37005 MR1739253 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. 1004.34033 MR1856591 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. 1004.34033 MR1856591 C. Zhang and H. Yao, “Converse problems of Fourier expansion and their applications,”

*Nonlinear Analysis A*, vol. 56, no. 5, pp. 761–779, 2004. 1044.42006 MR2036790 C. Zhang and H. Yao, “Converse problems of Fourier expansion and their applications,”*Nonlinear Analysis A*, vol. 56, no. 5, pp. 761–779, 2004. 1044.42006 MR2036790 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. 0872.35007 MR1384610 10.1007/BF02247885 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. 0872.35007 MR1384610 10.1007/BF02247885 C. Corduneanu,

*Almost čommentComment on ref. [28b*: We split this reference to ?,28b?]. Please check.Periodic Functions}, Interscience Publishers, Chelsea, NY, USA, 2nd edition, 1989. C. Corduneanu,*Almost čommentComment on ref. [28b*: We split this reference to ?,28b?]. Please check.Periodic Functions}, Interscience Publishers, Chelsea, NY, USA, 2nd edition, 1989. 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. MR1726918 10.1006/jdeq.1999.3651 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. MR1726918 10.1006/jdeq.1999.3651 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. 1127.35085 MR2346903 10.1016/j.jmaa.2007.01.098 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. 1127.35085 MR2346903 10.1016/j.jmaa.2007.01.098 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. 1207.35283 MR2753238 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. 1207.35283 MR2753238 C. Zhang and F. Yang, “Remotely almost periodic solutions of parabolic inverse problems,”

*Nonlinear Analysis A*, vol. 65, no. 8, pp. 1613–1623, 2006. 1106.35142 MR2248688 C. Zhang and F. Yang, “Remotely almost periodic solutions of parabolic inverse problems,”*Nonlinear Analysis A*, vol. 65, no. 8, pp. 1613–1623, 2006. 1106.35142 MR2248688 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. 1166.35303 MR2417489 10.1007/s11425-008-0006-2 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. 1166.35303 MR2417489 10.1007/s11425-008-0006-2 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. MR1243878 0787.34002 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. MR1243878 0787.34002