On a nonlinear parabolic equation with fractional Laplacian and integral conditions

References

• Jumarhon B, McKee S. On a heat equation with nonlinear and nonlocal boundary conditions. J Math Anal Appl. 1995;190:806–820.
   Web of Science ®Google Scholar
• Jumarhon B, Pidcock M. On a nonlinear Volterra integro-differential equation with a weakly singular kernel. Zamm Z Angew Math Mech. 1996;76(6):357–360.
   Web of Science ®Google Scholar
• Jumarhon B, McKee S. Product integration methods for solving a system of nonlinear Volterra integral equations. J Comput Appl Math. 1996;67(6):24–41.
   Google Scholar
• Lubashevski IA, Gafiichuk VV. A mathematical model of an ideal hierarchically organized life support system of distributed living media. Dokl Akad Nauk. 1996;351(5):611–613.
   Web of Science ®Google Scholar
• Nane E, Tuan NH. Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data. SIAM/ASA J Uncertain Quantif. 2018;6(1):302–338.
   Web of Science ®Google Scholar
• Koba H, Matsuoka H. Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian. Analysis. 2015;35(1):47–57.
   Google Scholar
• Metzler R, Klafter J. The random Walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339(1):1–77.
   Web of Science ®Google Scholar
• Constantin P. Euler equations, Navier–Stokes equations and turbulence. In: Cannone M., Miyakawa T. (eds) Mathematical foundation of turbulent viscous flows. Berlin: Springer; 2006. p. 1–43. (Lecture notes in mathematics; 1871).
   Google Scholar
• Vaquero JM, Sajavicius S. The two-level finite difference schemes for the heat equation with nonlocal initial condition. Appl Math Comput. 2019;342:166–177.
   Web of Science ®Google Scholar
• Avalishvili G, Avalishvili M. On nonclassical problems for first-order evolution equations. Georgian Math J. 2011;18(3):441–463. doi:10.1515/GMJ.2011.0028
   Web of Science ®Google Scholar
• Dokuchaev N. On recovering parabolic diffusions from their time-averages. Calc Var Partial Differ Equ. 2019;58(1), Paper No. 27, 14 pp.
   PubMed Web of Science ®Google Scholar
• Volodymyr SI, Nytrebych ZM, Pukach PY. Boundary-value problems with integral conditions for a system of Lamé equations in the space of almost periodic functions. Electron J Differ Equ. 2016;2016(304):1–12.
   Google Scholar
• Pulkina SL, Savenkova AE. A problem with a nonlocal with respect to time condition for multidimensional hyperbolic equations. Russ Math. 2016;60(10):33–43.
   Web of Science ®Google Scholar
• Shelukhin VV. A variational principle in problems that are nonlocal with respect to time for linear evolution equations. Sibirsk. Mat. Zh.. 1993;34(2). 191–207, 230, 236 (Russian); English transl., Siberian Math J. 34 (1993), no. 2, 369–384 (1994).
   Web of Science ®Google Scholar
• Tuan NH, Caraballo T. On initial and terminal value problems for fractional nonclassical diffusion equations. Proc Am Math Soc. 2020;149(1):1. doi:10.1090/proc/15131
   Web of Science ®Google Scholar
• A Kirsch. An introduction to the mathematical theory of inverse problems. 2nd ed. New York (NY): Springer; 2011. (Applied mathematical sciences; vol. 120).
   Google Scholar
• Ngoc TB, O'Regan NHTD. Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems. Commun Nonlinear Sci Numer Simul. 2019;78:104882, 13 pp.
   Web of Science ®Google Scholar
• Tuan NH, Zhou Y, Thach TN, et al. Initial inverse problem for the nonlinear fractional Rayleigh–Stokes equation with random discrete data. Commun Nonlinear Sci Numer Simul. 2019;78:104873, 18 pp.
   Web of Science ®Google Scholar
• Ngoc TB, Kian Y, Tuan NH. Inverse initial problem for fractional reaction-diffusion equation with nonlinearities. Preprint 2019. arXiv:1910.09006.
   Google Scholar
• Ngoc TB, Tuan NH, Caraballo T, et al. Existence and regularity results for terminal value problem for nonlinear super-diffusive fractional wave equations. Preprint 2019. arXiv:1910.00326.
   Google Scholar
• Tran N, Au VV, Zhou Y, et al. On a final value problem for fractional reaction-diffusion equation with Riemann–Liouville fractional derivative. Math Methods Appl Sci. 2019;43: 3086–3098. doi:10.1002/mma.6103
   Web of Science ®Google Scholar
• Triet NA, Au VV, Long LD, et al. Regularization of a terminal value problem for time fractional diffusion equation. Math Methods Appl Sci. 2020;43(6):3850–3878.
   Web of Science ®Google Scholar
• Shiri B, Wu GC, Baleanu D. Collocation methods for terminal value problems of tempered fractional differential equations. Appl Numer Math. 2020;156:385–395.
   Web of Science ®Google Scholar
• Tuan NH, Baleanu D, Thach TN, et al. Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data. J Comput Appl Math. 2020;376:112883, 25 pp.
   Web of Science ®Google Scholar
• Ashyralyev A, Ozturk E. On Bitsadze–Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: well-posedness. Appl Math Comput. 2012;219(3):1093–1107.
   Web of Science ®Google Scholar
• Kumar S, Baleanu D. Numerical solution of two-dimensional time fractional cable equation with Mittag–Leffler kernel. Math Methods Appl Sci. 2020;43(15):8348–8362.
   Web of Science ®Google Scholar
• Avalishvili G, Avalishvili M, Gordeziani D. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl Math Lett. 2011;24(4):566–571.
   Web of Science ®Google Scholar