Abstract
We consider nonlocal in time semilinear subdiffusion equations on a bounded domain, where the kernel in the integro-differential operator belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of fractional dynamics. The elliptic operator in the equation is given in divergence form with bounded measurable coefficients. We prove a well-posedness result in the setting of bounded weak solutions and study the stability and instability of the zero function in the special case where the nonlinearity vanishes at 0. We also establish a blowup result for positive convex and superlinear nonlinearities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahmad, B.; Alhothuali, M. S.; Alsulami, H. H.; Kirane, M.; Timoshin, S.: On a time fractional reaction diffusion equation. Appl. Math. Comput. 257 (2015), 199–204.
Arendt, W.; Prüss, J.: Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations. SIAM J. Appl. Math. 23 (1992), 412–448.
Clément, Ph.; Londen, S.-O.; Simonett, G.: Quasilinear evolutionary equations and continuous interpolation spaces. J. Differential Equations 196 (2004), 418–447.
Clément, Ph.; Nohel, J.A.: Abstract linear and nonlinear Volterra equations preserving positivity. SIAM J. Math. Anal. 10 (1979), 365–388.
Clément, Ph.; Nohel, J.A.: Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J. Math. Anal. 12 (1981), 514–534.
Clément, Ph.; Prüss, J.: Completely positive measures and Feller semigroups. Math. Ann. 287 (1990), 73–105.
Clément, Ph.; Prüss, J.: Global existence for a semilinear parabolic Volterra equation. Math. Z. 209 (1992), 17–26.
Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27 (1990), 309–321.
Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Springer, 1977.
Gripenberg, G.: An abstract nonlinear Volterra equation. Israel J. Math. 34 (1979), 198–212.
Gripenberg, G.: Volterra integro-differential equations with accretive nonlinearity. J. Differ. Eq. 60 (1985), 57–79.
Gripenberg, G.; Londen, S.-O.; Staffans, O.: Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990.
Gurtin, M. E.; Pipkin, A. C.: A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31 (1968), 113–126.
Kaplan, S.: On the growth of solutions of quasi-linear parabolic equations. Comm. Pure Appl. Math. 16 (1963), 305–330.
Kato, N.: Linearized stability for semilinear Volterra integral equations. Differential Integral Equations 8 (1995), 201–212.
Kemppainen, J.; Siljander, J.; Vergara, V.; Zacher, R.: Decay estimates for time-fractional and other non-local in time subdiffusion equations in \({\mathbb{R}}^d\). Math. Ann. (2016). doi:10.1007/s00208-015-1356-z
Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations. Elsevier, 2006.
Kirane, M.; Laskri, Y.; Tatar, N.-e.: Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives. J. Math. Anal. Appl. 312 (2005), 488–501.
Kochubei, A. N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340 (2008), 252–281.
Kochubei, A. N.: General fractional calculus, evolution equations, and renewal processes. Integr. Equ. Oper. Theory 71 (2011), 583–600.
Malolepszy, T.; Okrasiński, W.: Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity. Appl. Math. Lett. 21 (2008), 307–312.
Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.
Nakagawa, J.; Sakamoto, K.; Yamamoto, M.: Overview to mathematical analysis for fractional diffusion equations – new mathematical aspects motivated by industrial collaboration. J. Math-for-Ind. 2A (2010), 99–108.
Nunziato, J. W.: On heat conduction in materials with memory. Quart. Appl. Math. 29 (1971), 187–204.
Olmstead, W. E.; Roberts, C. A.: Thermal blow-up in a subdiffusive medium. SIAM J. Appl. Math. 69 (2008), 514–523.
Prüss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
Prüss, J.; Vergara, V.; Zacher, R.: Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete Contin. Dyn. Syst. 26 (2010), 625–647.
Quittner, P.; Souplet, Ph.: Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts. Birkhäuser, Basel, 2007.
Roberts, C. A.; Lasseigne, D. G.; Olmstead, W. E.: Volterra equations which model explosion in a diffusive medium. J. Integral Equations Appl. 5 (1993), 531–546.
Schneider, W. R.; Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144.
Uchaikin, V. V.: Fractional derivatives for physicists and engineers. Volume I Background and Theory. Nonlinear Physical Science, Springer, Heidelberg, 2013.
Vergara, V.; Zacher, R.: Lyapunov functions and convergence to steady state for differential equations of fractional order. Math. Z. 259 (2008), 287–309.
Vergara, V.; Zacher, R.: Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods. SIAM J. Math. Anal. 47 (2015), 210–239.
Zacher, R.: A weak Harnack inequality for fractional evolution equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), 903–940.
Zacher, R.: Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients. J. Math. Anal. Appl. 348 (2008), 137–149.
Zacher, R.: Global strong solvability of a quasilinear subdiffusion problem. J. Evol. Equ. 12 (2012), 813–831.
Zacher, R.: Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations. J. Evol. Equ. 5 (2005), 79–103.
Zacher, R.: Quasilinear parabolic integro-differential equations with nonlinear boundary conditions. Differential Integral Equations 19 (2006), 1129–1156.
Zacher, R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj 52 (2009), 1–18.
Zeidler, E.: Nonlinear functional analysis and its applications. I: Fixed-point theorems. Springer-Verlag, New York, 1986.
Author information
Authors and Affiliations
Corresponding author
Additional information
V. Vergara was partially supported by FONDECYT Grant 1150230.
R. Zacher was partially supported by a Heisenberg fellowship of the German Research Foundation (DFG), GZ Za 547/3-1.
Dedicated to Jan Prüss on the occasion of his 65th birthday.
Rights and permissions
About this article
Cite this article
Vergara, V., Zacher, R. Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations. J. Evol. Equ. 17, 599–626 (2017). https://doi.org/10.1007/s00028-016-0370-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-016-0370-2