On initial and terminal value problems for fractional nonclassical diffusion equations
Authors:
Nguyen Huy Tuan and Tomás Caraballo
Journal:
Proc. Amer. Math. Soc. 149 (2021), 143-161
MSC (2010):
Primary 26A33, 35B65, 35R11
DOI:
https://doi.org/10.1090/proc/15131
Published electronically:
June 11, 2020
Uncorrected version:
Original version posted June 11, 2020
Corrected version:
This paper was updated due to incomplete funding information at the time of electronic publication.
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case and show the ill-posedness in the sense of Hadamard in the case
. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in
norm is first established.
- [1] Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- [2] Yang Cao, Jingxue Yin, and Chunpeng Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations 246 (2009), no. 12, 4568–4590. MR 2523294, https://doi.org/10.1016/j.jde.2009.03.021
- [3] A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl. 310 (2005), no. 2, 557–578. MR 2022944, https://doi.org/10.1016/j.jmaa.2005.02.024
- [4] Hang Ding and Jun Zhou, Global existence and blow-up for a mixed pseudo-parabolic 𝑝-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl. 478 (2019), no. 2, 393–420. MR 3979113, https://doi.org/10.1016/j.jmaa.2019.05.018
- [5] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Rational Mech. Anal. 49 (1972/73), 57–78. MR 330774, https://doi.org/10.1007/BF00281474
- [6] Yijun He, Huaihong Gao, and Hua Wang, Blow-up and decay for a class of pseudo-parabolic 𝑝-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl. 75 (2018), no. 2, 459–469. MR 3765862, https://doi.org/10.1016/j.camwa.2017.09.027
- [7] Lingyu Jin, Lang Li, and Shaomei Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl. 73 (2017), no. 10, 2221–2232. MR 3641746, https://doi.org/10.1016/j.camwa.2017.03.005
- [8] Tsuan Wu Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440–453. MR 264231, https://doi.org/10.2969/jmsj/02130440
- [9] Nguyen Huy Tuan and Dang Duc Trong, A nonlinear parabolic equation backward in time: regularization with new error estimates, Nonlinear Anal. 73 (2010), no. 6, 1842–1852. MR 2661365, https://doi.org/10.1016/j.na.2010.05.019
- [10] Tuan Nguyen Huy, Mokhtar Kirane, Bessem Samet, and Van Au Vo, A new Fourier truncated regularization method for semilinear backward parabolic problems, Acta Appl. Math. 148 (2017), 143–155. MR 3621295, https://doi.org/10.1007/s10440-016-0082-1
- [11] Hongwei Zhang, Jun Lu, and Qingying Hu, Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Appl. 68 (2014), no. 12, 1787–1793. MR 3283367, https://doi.org/10.1016/j.camwa.2014.10.012
- [12] Nguyen Huy Tuan, Doan Vuong Nguyen, Vo Van Au, and Daniel Lesnic, Recovering the initial distribution for strongly damped wave equation, Appl. Math. Lett. 73 (2017), 69–77. MR 3659910, https://doi.org/10.1016/j.aml.2017.04.014
- [13] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. MR 2426856, https://doi.org/10.1515/JIIP.2008.019
- [14] Pan Dai, Chunlai Mu, and Guangyu Xu, Blow-up phenomena for a pseudo-parabolic equation with 𝑝-Laplacian and logarithmic nonlinearity terms, J. Math. Anal. Appl. 481 (2020), no. 1, 123439, 27. MR 4008539, https://doi.org/10.1016/j.jmaa.2019.123439
- [15] José M. Arrieta and Alexandre N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc. 352 (2000), no. 1, 285–310. MR 1694278, https://doi.org/10.1090/S0002-9947-99-02528-3
- [16] Tahir Bachar Issa and Wenxian Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 2, 926–973. MR 3648963, https://doi.org/10.1137/16M1092428
- [17] Tahir Bachar Issa and Wenxian Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, J. Dynam. Differential Equations 31 (2019), no. 4, 1839–1871. MR 4028556, https://doi.org/10.1007/s10884-018-9686-7
- [18] Tahir Bachar Issa and Wenxian Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, J. Dynam. Differential Equations 31 (2019), no. 4, 1839–1871. MR 4028556, https://doi.org/10.1007/s10884-018-9686-7
- [19] Tomás Caraballo, Antonio M. Márquez-Durán, and Felipe Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 5, 1817–1833. MR 3627130, https://doi.org/10.3934/dcdsb.2017108
- [20] Tomás Caraballo, Antonio M. Márquez-Durán, and Felipe Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 25 (2015), no. 14, 1540021, 11. MR 3448562, https://doi.org/10.1142/S0218127415400210
- [21] E. M. Bonotto, M. C. Bortolan, T. Caraballo, and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations 262 (2017), no. 6, 3524–3550. MR 3592649, https://doi.org/10.1016/j.jde.2016.11.036
- [22] Renhai Wang, Yangrong Li, and Bixiang Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst. 39 (2019), no. 7, 4091–4126. MR 3960498, https://doi.org/10.3934/dcds.2019165
- [23] Bruno de Andrade, On the well-posedness of a Volterra equation with applications in the Navier-Stokes problem, Math. Methods Appl. Sci. 41 (2018), no. 2, 750–768. MR 3745344, https://doi.org/10.1002/mma.4642
- [24] Bruno de Andrade and Arlúcio Viana, Abstract Volterra integrodifferential equations with applications to parabolic models with memory, Math. Ann. 369 (2017), no. 3-4, 1131–1175. MR 3713537, https://doi.org/10.1007/s00208-016-1469-z
- [25] Bruno de Andrade and Arlúcio Viana, Integrodifferential equations with applications to a plate equation with memory, Math. Nachr. 289 (2016), no. 17-18, 2159–2172. MR 3583262, https://doi.org/10.1002/mana.201500205
- [26] Bruno de Andrade, Alexandre N. Carvalho, Paulo M. Carvalho-Neto, and Pedro Marín-Rubio, Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 439–467. MR 3408831, https://doi.org/10.12775/TMNA.2015.022
- [27] Andrew B. Ferrari and Edriss S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations 23 (1998), no. 1-2, 1–16. MR 1608488, https://doi.org/10.1080/03605309808821336
- [28] Renhai Wang, Lin Shi, and Bixiang Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on ℝ^{ℕ}, Nonlinearity 32 (2019), no. 11, 4524–4556. MR 4022971, https://doi.org/10.1088/1361-6544/ab32d7
- [29] R. Wang, Y. Li, B. Wang, Bi-spatial pullback attractors of fractional nonclassical diffusion equation- s on unbounded domains with (p, q)-growth nonlinearities, Applied Mathematics and Optimization, (2020), doi.org/10.1007/s00245-019-09650-6.
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26A33, 35B65, 35R11
Retrieve articles in all journals with MSC (2010): 26A33, 35B65, 35R11
Additional Information
Nguyen Huy Tuan
Affiliation:
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam; and Vietnam National University, Ho Chi Minh City, Vietnam
MR Author ID:
777405
Email:
nhtuan@hcmus.edu.vn
Tomás Caraballo
Affiliation:
Departamento de Ecuaciones Diferenciales y Análisis Numérico C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain
Email:
caraball@us.es
DOI:
https://doi.org/10.1090/proc/15131
Keywords:
Fractional nonclassical diffusion equation,
well-posedness,
ill-posedness,
regularity estimates,
regularization and error estimate.
Received by editor(s):
November 7, 2019
Received by editor(s) in revised form:
January 15, 2020
Published electronically:
June 11, 2020
Additional Notes:
This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09.
The research of the second author was partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00.
Communicated by:
Wenxian Shen
Article copyright:
© Copyright 2020
American Mathematical Society