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A DIFFUSIVE LOGISTIC EQUATION WITH MEMORY IN BESSEL POTENTIAL SPACES

Part of: Parabolic equations and systems Representations of solutions

Published online by Cambridge University Press:  16 June 2015

ALEJANDRO CAICEDO  and
ARLÚCIO VIANA
ALEJANDRO CAICEDO
Affiliation:
Departamento de Matemática, Universidade Federal de Sergipe, Avenue Vereador Olímpio Grande, Itabaiana-SE, Brazil email [email protected]
ARLÚCIO VIANA*
Affiliation:
Departamento de Matemática, Universidade Federal de Sergipe, Avenue Vereador Olímpio Grande, Itabaiana-SE, Brazil email [email protected]
*
✉[email protected]
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Abstract

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This paper is devoted to the study of the local existence, uniqueness, regularity, and continuous dependence of solutions to a logistic equation with memory in the Bessel potential spaces.

Keywords

partial integrodifferential equationBessel potential spaceslogistic equationmemory effectlocal well-posedness

MSC classification

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35D30: Weak solutions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 251 - 258
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics, A Series of Monographs and Textbooks, 65 (Academic Press, New York, 1975).Google Scholar
Amann, H., Linear and Quasilinear Parabolic Problems. Abstract Linear Theory: I, Monographs in Mathematics, 89 (Birkhäuser Verlag, Boston, 1995).CrossRefGoogle Scholar
Amann, H., ‘Navier–Stokes equations with nohomogeneous Dirichlet data’, J. Nonlinear Phys. 10(1) (2003), 1–11.CrossRefGoogle Scholar
Bothe, D., Köhne, M. and Prüss, J., ‘On a class of energy preserving boundary conditions for incompressible Newtonian flows’, SIAM J. Math. Anal. 45(6) (2013), 3768–3822.CrossRefGoogle Scholar
Cantrell, R. S. and Cosner, C., Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology (Wiley, Chichester, 2003).Google Scholar
Cantrell, R. S. and Cosner, C., ‘Density dependent behavior at habitat boundaries and the Allee effect’, Bull. Math. Biol. 69 (2007), 2339–2360.CrossRefGoogle ScholarPubMed
Cushing, J. M., ‘Volterra integrodifferential equations in population dynamics’, Math. Biol. 80 (1979), 81–148.Google Scholar
Du, Y., Peng, R. and Polácik, P., ‘The parabolic logistic equation with blow-up initial and boundary values’, J. Anal. Math. 118 (2012), 297–316.CrossRefGoogle Scholar
Dyson, J., Villella-Bressan, R. and Webb, G., ‘Asymptotic behaviour of solutions to abstract logistic equations’, Math. Biosci. 206 (2007), 216–232.CrossRefGoogle ScholarPubMed
Feng, W. and Lu, X., ‘Asymptotic periodicity in diffusive logistic equations with discrete delays’, Nonlinear Anal. 26 (1996), 171–178.CrossRefGoogle Scholar
Goddard II, J., Shivaji, R. and Lee, E. K., ‘Diffusive logistic equation with non-linear boundary conditions’, J. Math. Anal. Appl. 375 (2011), 365–370.CrossRefGoogle Scholar
Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74 (Kluwer Academic Publishers Group, Dordrecht, 1992).CrossRefGoogle Scholar
Hadeler, K. P., ‘Diffusion equations in biology’, Math. Biol. 80 (1979), 149–177.Google Scholar
Harris, S., ‘Diffusive logistic population growth with immigration’, Appl. Math. Lett. 18 (2005), 261–265.CrossRefGoogle Scholar
Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (Springer, Berlin, 1980).Google Scholar
Oruganti, S., Shi, J. and Shivaji, R., ‘Diffusive logistic equations with constant yield harvesting, I: Steady solutions’, Trans. Amer. Math. Soc. 354(9) (2002), 3601–3619.CrossRefGoogle Scholar
Rodrigues, J. F. and Tavares, H., ‘Increasing powers in a degenerate parabolic logistic equation’, Chin. Ann. Math. 34B(2) (2013), 277–294.CrossRefGoogle Scholar
Seifert, G., ‘‘Almost periodic solutions for delay logistic equations with almost periodic time dependence’’, Differ. Integral Equ. 9(2) (1996), 335–342.Google Scholar
Schiaffino, A., ‘On a diffusion Volterra equation’, Nonlinear Anal. 3(5) (1979), 595–600.CrossRefGoogle Scholar
Schumacher, K., ‘The instationary Stokes equations in weighted Bessel-potential spaces’, J. Evol. Equ. 9 (2009), 1–36.CrossRefGoogle Scholar
Steiger, O., ‘Navier–Stokes equations with first order boundary conditions’, J. Math. Fluid Mech. 8 (2006), 456–481.CrossRefGoogle Scholar
Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978).Google Scholar
Viana, A., ‘Local well-posedness for a Lotka–Volterra system in Besov spaces’, Comput. Math. Appl. 69 (2015), 667–674.CrossRefGoogle Scholar
Yamada, Y., ‘On a certain class of semilinear Volterra diffusion equations’, J. Math. Anal. Appl. 88 (1982), 433–451.CrossRefGoogle Scholar
Yang, X., Wang, W. and Shen, J., ‘Permanence of a logistic type impulsive equation with infinite delay’, Appl. Math. Lett. 24 (2011), 420–427.CrossRefGoogle Scholar
Zhao, C., Debnath, L. and Wang, K., ‘Positive periodic solutions of a delayed model in population’, Appl. Math. Lett. 16 (2003), 561–565.CrossRefGoogle Scholar