Tokyo Journal of Mathematics

Asymptotic Behavior of Solutions for Semilinear Volterra Diffusion Equations with Spatial Inhomogeneity and Advection

Yoshio YAMADA and Yusuke YOSHIDA

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This paper is concerned with semilinear Volterra diffusion equations with spatial inhomogeneity and advection. We intend to study the effects of interaction among diffusion, advection and Volterra integral under spatially inhomogeneous environments. Since the existence and uniqueness result of global-in-time solutions can be proved in the standard manner, our main interest is to study their asymptotic behavior as $t\to \infty$. For this purpose, we study the related stationary problem by the monotone method and establish some sufficient conditions on the existence of a unique positive solution. Its global attractivity is also studied with use of a suitable Lyapunov functional.

Article information

Tokyo J. Math., Volume 39, Number 1 (2016), 271-292.

First available in Project Euclid: 30 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35J61: Semilinear elliptic equations 35K57: Reaction-diffusion equations 35R09: Integro-partial differential equations [See also 45Kxx] 92D25: Population dynamics (general)


YOSHIDA, Yusuke; YAMADA, Yoshio. Asymptotic Behavior of Solutions for Semilinear Volterra Diffusion Equations with Spatial Inhomogeneity and Advection. Tokyo J. Math. 39 (2016), no. 1, 271--292. doi:10.3836/tjm/1459367268.

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  • S. Ahmad and M. R. M. Rao, Stability of Volterra diffusion equations with time delays, Appl. Math. Comput. 90 (1998), 143–154.
  • H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125–146.
  • F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canad. Appl. Math. Quart. 3 (1995), 379–397.
  • H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), 1–29.
  • A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal. 89 (2010), 983–1004.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011.
  • K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eignevalue problem with indefinite weight function, J. Math. Anal. Appl. 75 (1980), 112–120.
  • R. F. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 293–318.
  • R. F. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol. 29 (1991), 315–338.
  • R. F. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003.
  • C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl. 277 (2003), 489–503.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics Vol. 840, Springer-Verlag, Berlin-New York, 1981.
  • H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac. 34 (1991), 475–494.
  • M. Iida, Exponentially asymptotic stability for a certain class of semilinear Volterra diffusion equations, Osaka J. Math. 28 (1991), 411–440.
  • K-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations 250 (2011), 161–181.
  • H. Liu and H. Yu, Entropy/energy stable schemes for evolutionary dispersal models, J. Comput. Phys. 256 (2014), 656–677.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences Vol. 44, Springer-Verlag, New York, 1983.
  • M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
  • R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal. 16 (1985), 135–142.
  • F. Rothe, Uniform bounds from bounded $L_{p}$-functionals in reaction-diffusion equations, J. Differential Equations 45 (1982), 207–233.
  • F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics Vol. 1072, Springer-Verlag, Berlin, 1984.
  • D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979–1000.
  • A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal. 3 (1979), 595–600.
  • A. Schiaffino and A. Tesei, Monotone methods and attractivity results for Volterra integro-partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 135–142.
  • J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Grundlehren der Mathematischen Wissenschaften Vol. 258, Springer-Verlag, New York, 1994.
  • A. Tesei, Stability properties for partial Volterra integro-differential equations, Ann. Mat. Pura Appl. 126 (1980), 103–115.
  • Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl. 88 (1982), 433–457.
  • Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differential Equations 52 (1984), 295–326.
  • Y. Yamada, Asymptotic behavior of solutions for semilinear Volterra diffusion equations, Nonlinear Anal. 21 (1993), 227–239.