Globally attractive mild solutions for non-local in time subdiffusion equations of neutral type
Keywords
Attractive mild solutions, non-local in time equations, neutral type equations, integral Volterra equationsAbstract
We prove the existence of at least one globally attractive mild solution to the equation $$ \partial_t (b*[x-h(\cdot,x(\cdot))])(t) + A(x(t) - h(t,x(t))) = f(t,x(t)), \quad t\geq 0, $$under the assumption, among other hypothesis, that $A$ is an almost sectorial operator on a Banach space $X$ and the kernel $b$ belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of time-fractional evolution equations of neutral type.References
J.O. Alzabut and T. Abdeljawad, On existence of a globally attractive periodic solution of impulsive delay logarithmic population model, Appl. Math. Comput. 198 (2008), no. 1, 463–469.
J. Banaś and K. Goebel, Measures of Noncompactness in Banach space, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel–Dekker, New York, 1980.
M.V. Bartuccelli, J.H.B. Deane and G. Gentile, Guido Globally and locally attractive solutions for quasi-periodically forced systems, J. Math. Anal. Appl. 328 (2007), no. 1, 699–714.
Ph. Clément and J.A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal. 12 (1981), 514–534.
B. de Andrade and A. Viana, Abstract Volterra integrodifferential equations with applications to parabolic models with memory, Math. Ann. 369 (2017), no. 3–4, 1131–1175.
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for timefractional and other non-local in time subdiffusion equations in R, Math. Ann. 366 (2016), no. 3–4, 941–979.
A.N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), 252–281.
J.W. Li and S. Cheng, Globally attractive periodic solution of a perturbed functional differential equation, J. Comput. Appl. Math. 193 (2006), no. 2, 652–657.
J. Liang, S-H. Yan, F. Li, T-W. Huang, On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equation., Adv. Differential Equations 2012 (2012), 1–27.
F. Periago and B. Straub, A functional calculus for almost sectorial operators an applications to abstract evolution equations. J. Evol. Equ. 2 (2002), 41–68.
A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series, vol. 1, Taylor and Francis, London, 2002.
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993.
J. Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), no. 2, 317–344.
X.H. Tang, Asymptotic behavior of delay differential equations with instantaneously terms, J. Math. Anal. Appl. 302 (2005), no. 2, 342–359.
V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal. 47 (2015), no. 1, 210–239.
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