Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

Nguyen Duc Phuong¹ ; Nguyen Anh Tuan² ; Devendra Kumar³ ; Nguyen Huy Tuan²

¹ Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam.
² Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam.
³ Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 27.

Résumé

In this paper, we investigate a initial value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace operator of order 0 ν ≤ 1 and the nonlinear memory source term. For 0 ν 1, the problem will be considered on a bounded domain of ℝd. By some Sobolev embeddings and the properties of the Mittag-Leffler function, we will give some results on the existence and the uniqueness of mild solution for problem (1.1) below. When ν = 1, we will introduce some Lp − Lq estimates, and based on them we derive the global existence of a mild solution in the whole space ℝd.

Détail

DOI : 10.1051/mmnp/2021015

Bibliographie

[1] B. Andrade, A. Viana Abstract Volterra integrodifferential equations with applications to parabolic models with memory Math. Ann 2017 1131 1175

[2] J.M. Arrieta, A.N. Carvalho Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations Trans. Am. Math. Soc 2000 285 310

[3] D. Baleanu, B. Shiri, H.M. Srivastava, M. Al Qurashi A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel Adv. Diff. Equ 2018 353

[4] D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative Chaos Solitons Fract 2020 109705

[5] D. Baleanu, A. Jajarmi, S.S. Sajjadi, J.H. Asad The fractional features of a harmonic oscillator with position-dependent mass Commun. Theor. Phys 2020 055002 55010

[6] M.Kh. Beshtokov Boundary value problems for a pseudo-parabolic equation with the Caputo fractional derivative Differ. Equ 2019 884 893

[7] Y. Cao, J. Yin, C. Wang Cauchy problems of semilinear pseudo-parabolic equations J. Diff. Equ 2009 4568 4590

[8] Y. Cao, J. Yin, C. Wang Cauchy problems of semilinear pseudo-parabolic equations J. Diff. Equ 2009 4568 4590

[9] Y. Chen, H. Gao, M. Garrido-Atienza, B. Schmalfuß Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems. Disc. Cont. Dyn. Syst. Ser. A 2014 79 98

[10] W. Chen, C. Li Maximum principles for the fractional p-Laplace and symmetry of solutions Adv. Math 2018 735 758

[11] H. Chen, S. Tian Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity J. Differ. Equ 2015 4424 4442

[12] E. Di Nezza, G. Palatucci, E. Valdinoci Hitchhiker’s guide to the fractional Sobolev spaces Bull. Sci. Math 2012 521 573

[13] R. Gorenfloo, A.A. Kilbas and F. Mainardi, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).

[14] A. Jajarmi, A. Yusuf, D. Baleanu, M. Inc A new fractional HRSV model and its optimal control: a non-singular operator approach Physica A 2020 123860

[15] A Jajarmi, D. Baleanu On the fractional optimal control problems with a general derivative operator Asian J. Cont 2019

[16] S. Ji, J. Yin, Y. Cao Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity J. Differ. Equ 2016 5446 5464

[17] L. Jin, L. Li, S. Fang The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation Comput. Math. Appl 2017 2221 2232

[18] E.D. Khiabani, H. Ghaffarzadeh, B. Shiri, J. Katebi Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models J. Vibr. Contr 2020 1445 1462

[19] W. Lian, J. Wang, R. Xu Global existence and blow up of solutions for pseudo-parabolic equation with singular potential J. Diff. Equ 2020 4914 4959

[20] Y. Liu, W. Jiang, F. Huang Asymptotic behaviour of solutions to some pseudo-parabolic equations Appl. Math. Lett 2012 111 114

[21] F. Mainardi, A. Mura, G. Pagnini The M-Wright function in time-fractional diffusion processes: a tutorial survey Int. J. Differ. Equ 2010

[22] B.B. Mandelbrot, J.W.V Ness Fractional Brownian motions, fractional noises and applications SIAM Rev 1968 422 437

[23] V. Padron Effect of aggregation on population recovery modeled by a forward-backward pseudo-parabolic equation Trans. Amer. Math. Soc 2004 2739 2756

[24] J.E.M. Rivera, L.H. Fatori Smoothing effect and propagations of singularities for viscoelastic plates J. Math. Anal. Appl 1997 397 427

[25] M.K. Sadabad, A.J. Akbarfam, B. Shiri A numerical study of eigenvalues and eigenfunctions of fractional Sturm-Liouville problems via Laplace transform Indian J. Pure Appl. Math 2020 857 868

[26] S.S. Sajjadi, D. Baleanu, A. Jajarmi, H.M. Pirouz A new adaptive synchronization and hyperchaos control of a biological snap oscillator Chaos Solitons Fract 2020 13

[27] B. Shiri, G.-C. Wu, D. Baleanu Collocation methods for terminal value problems of tempered fractional differential equations Appl. Numer. Math 2020 385 395

[28] B. Shiri, D. Baleanu Numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order Results Nonlinear Anal 2019 160 168

[29] B. Shiri, D. Baleanu System of fractional differential algebraic equations with applications Chaos Solitons Fractals 2019 203 212

[30] R.E. Showalter, T.W. Ting Pseudo-parabolic partial differential equations SIAM J. Math. Anal 1970 1 26

[31] N.H. Tuan, V.V. Au, R. Xu Semilinear Caputo time-fractional pseudo-parabolic equations Commun. Pure Appl. Anal 2021 583 621

[32] R. Xu, J. Su Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations J. Funct. Anal 2013 2732 2763