Abstract
This paper is concerned with the blow-up phenomena and global existence of a fractional nonlinear reaction-diffusion equation with a non-local source term. Under sufficient conditions on the weight function a(x) and when the initial data is small enough, the global existence of solutions is proved using the comparison principle. We establish a finite time blow-up of the solution with large initial data by converting the fractional PDE into a simple ordinary differential inequality using the differential inequality technique. Moreover, by solving the obtained ordinary differential inequality, an upper bound of the blow-up time is also deduced.
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References
Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractional order equations. Differ. Equ. 46, 660–666 (2010). https://doi.org/10.1134/S0012266110050058
Allegretto, W., Fragnelli, G., Nistri, P., Papini, D.: Coexistence and optimal control problems for a degenerate predator-prey model. J. Math. Anal. Appl. 378, 528–540 (2011). https://doi.org/10.1016/j.jmaa.2010.12.036
Bebernes, J., Bressan, A.: Thermal behavior for a confined reactive gas. J. Differ. Equ. 44, 118–133 (1982). https://doi.org/10.1016/0022-0396(82)90028-6
Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory, Applied Mathematical Sciences. Springer, N.Y. (1989). https://doi.org/10.1007/978-1-4612-4546-9
Borwein, P., Erdlyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-0793-1
Calsina, À., Perelló, C., Saldaña, J.: Non-local reaction-diffusion equations modelling predator-prey coevolution. Publ. Mat. 38, 315–325 (1994). https://www.jstor.org/stable/43736486
Cao, J., Song, G., Wang, J., Shi, Q., Sun, S.: Blow-up and global solutions for a class of time fractional nonlinear reaction-diffusion equation with weakly spatial source. Appl. Math. Lett. 91, 201–206 (2019). https://doi.org/10.1016/j.aml.2018.12.020
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer. Berlin Heidelberg (2010). https://doi.org/10.1007/978-3-642-14573-5
Furter, J., Grinfeld, M.: Local vs. nonlocal interactions in population dynamics. J. Math. Biol. 27, 65–80 (1989). https://doi.org/10.1007/BF00276081
Hu, B.: Blow-up Theories for Semilinear Parabolic Equations. Lecture Notes in Mathematics. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-642-18460-4
Ivanauskas, F., Laurinavičius, V., Sapagovas, M., Nečiporenko, A.: Reaction-diffusion equation with nonlocal boundary condition subject to PID-controlled bio-reactor. Nonlinear Anal. Model. Control. 22, 261–272 (2017). https://doi.org/10.15388/NA.2017.2.8
Khramchenkov, M.G.: Dispersion and chemical reactions in porous media. Fluid Dyn. 36, 166–168 (2001). https://doi.org/10.1023/A:1018896114067
Ma, L., Fang, Z.B.: Blow-up analysis for a nonlocal reaction-diffusion equation with robin boundary conditions. Taiwan. J. Math. 21, 131–150 (2017). https://doi.org/10.11650/tjm.21.2017.7380
Ma, L.,Fang, Z.B.: Blow-up phenomena of solutions for a reaction-diffusion equation with weighted exponential nonlinearity. Comput. Math. Appl. 75, 2735–2745 (2018). https://doi.org/10.1016/j.camwa.2018.01.005
Manimaran, J., Shangerganesh, L.: Blow-up solutions of a time-fractional diffusion equation with variable exponents. Tbil. Math. J. 12, 149–157 (2019). https://doi.org/10.32513/tbilisi/1578020574
Pinasco, J.P.: Blow-up for parabolic and hyperbolic problems with variable exponents. Nonlinear Anal. 71, 1094–1099 (2009). https://doi.org/10.1016/j.na.2008.11.030
Tang, G., Li, Y., Yang, X.: Lower bounds for the blow-up time of the nonlinear nonlocal reaction diffusion problems in \(\cal{R}^N\)\((N \ge 3)\), Bound. Value Probl. 1, 1–5 (2014). https://doi.org/10.1186/s13661-014-0265-5
Tao, X., Fang, Z.B.: Blow-up phenomena for a nonlinear reaction-diffusion system with time dependent coefficients. Comput. Math. Appl. 74, 2520–2528 (2017). https://doi.org/10.1016/j.camwa.2017.07.037
Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier-Stokes equations and optimal control. Comput. Math. Appl. 6, 1016–1027 (2017). https://doi.org/10.1016/j.camwa.2016.07.007
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Saranya, R., Annapoorani, N. (2023). Blow-up Analysis and Global Existence of Solutions for a Fractional Reaction-Diffusion Equation. In: Sharma, R.K., Pareschi, L., Atangana, A., Sahoo, B., Kukreja, V.K. (eds) Frontiers in Industrial and Applied Mathematics. FIAM 2021. Springer Proceedings in Mathematics & Statistics, vol 410. Springer, Singapore. https://doi.org/10.1007/978-981-19-7272-0_6
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