On initial and terminal value problems for fractional nonclassical diffusion equations

Tuan, Nguyen Huy; Caraballo, Tomás

doi:10.1090/proc/15131

On initial and terminal value problems for fractional nonclassical diffusion equations
HTML articles powered by AMS MathViewer

by Nguyen Huy Tuan and Tomás Caraballo PDF

Proc. Amer. Math. Soc. 149 (2021), 143-161 Request permission

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 $0<\alpha \le 1$ and show the ill-posedness in the sense of Hadamard in the case $\alpha > 1$. 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 $L^q$ norm is first established.

References

Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
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, DOI 10.1016/j.jde.2009.03.021
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, DOI 10.1016/j.jmaa.2005.02.024
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, DOI 10.1016/j.jmaa.2019.05.018
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, DOI 10.1007/BF00281474
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, DOI 10.1016/j.camwa.2017.09.027
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, DOI 10.1016/j.camwa.2017.03.005
Tsuan Wu Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440–453. MR 264231, DOI 10.2969/jmsj/02130440
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, DOI 10.1016/j.na.2010.05.019
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, DOI 10.1007/s10440-016-0082-1
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, DOI 10.1016/j.camwa.2014.10.012
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, DOI 10.1016/j.aml.2017.04.014
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, DOI 10.1515/JIIP.2008.019
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, DOI 10.1016/j.jmaa.2019.123439
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, DOI 10.1090/S0002-9947-99-02528-3
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, DOI 10.1137/16M1092428
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, DOI 10.1007/s10884-018-9686-7
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, DOI 10.1007/s10884-018-9686-7
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, DOI 10.3934/dcdsb.2017108
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, DOI 10.1142/S0218127415400210
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, DOI 10.1016/j.jde.2016.11.036
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, DOI 10.3934/dcds.2019165
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, DOI 10.1002/mma.4642
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, DOI 10.1007/s00208-016-1469-z
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, DOI 10.1002/mana.201500205
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, DOI 10.12775/TMNA.2015.022
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, DOI 10.1080/03605309808821336
Renhai Wang, Lin Shi, and Bixiang Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\Bbb R^N$, Nonlinearity 32 (2019), no. 11, 4524–4556. MR 4022971, DOI 10.1088/1361-6544/ab32d7
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.