Global and local existence of solutions for nonlinear systems of time-fractional diffusion equations
Abstract: In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the existence of solutions global in time and the blow-up. Our approach involves the truncation of the nonlinear terms, which enables us to handle all local Lipschitz continuous nonlinear terms, provided their sum is less than or equal to zero. By employing a comparison principle for the corresponding linear system, we establish also the non-negativity of the nonlinear system.
- Adams R A 1975 Sobolev Spaces (New York: Academic Press)
- Bergh J and Löfström J 1976 Interpolation Spaces: An Introduction (Springer-Verlag: Berlin)
- Floridia G, Liu Y and Yamamoto M 2023 Blow-up in L1(Ω)superscript𝐿1ΩL^{1}(\Omega)italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω )-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms Advances in Nonlinear Analysis 12(1) 20230121
- Gorenflo R, Luchko Y and Yamamoto M 2015 Time-fractional diffusion equation in the fractional Sobolev spaces Fract. Calc. Appl. Anal. 18 799-820
- Klausmeier C A 1999 Regular and irregular patterns in semiarid vegetation Science 284(5421) 1826-1828
- Kubica A, Ryszewska K and Yamamoto M 2020 Theory of Time-fractional Differential Equations An Introduction (Tokyo: Springer)
- Li Z, Huang X and Liu Y 2023 Initial-boundary value problems for coupled systems of time-fractional diffusion equations Fractional Calculus and Applied Analysis 26(2) 533-566
- Luchko Y and Yamamoto M 2017 On the maximum principle for a time-fractional diffusion equation Fract. Calc. Appl. Anal. 20 1131-1145
- Luchko Y and Yamamoto M 2022 Comparison principles for the linear and semiliniar time-fractional diffusion equations with the Robin boundary condition arXiv preprint arXiv:2208.04606
- Pazy A 1983 Semigroups of Linear Operators and Applications to Partial Differential Equations (Berlin: Springer)
- Pierre M 2003 Weak solutions and supersolutions in L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT for reaction-diffusion systems J. Evol. Equ. 3 153-168
- Pierre M 2010 Global existence in reaction-diffusion systems with control of mass: a survey Milan Journal of Mathematics 78 417-455
- Sakamoto K and Yamamoto M 2011 Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems J. Math. Anal. Appl. 382 426-447
- Simon J 1986 Compact sets in the space Lp(0,T;B)superscript𝐿𝑝0𝑇𝐵L^{p}(0,T;B)italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( 0 , italic_T ; italic_B ) Annali di Matematica pura ed applicata 146 65-96
- Wang X, Shi J and Zhang G 2021 Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction. J. Math. Anal. Appl. 497(1) 124860
- Yamamoto M 2022 Fractional calculus and time-fractional differential equations: revisit and construction of a theory Mathematics 10 698
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