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Global existence for semilinear parabolic systems via Lyapunov type methods

Part of the Lecture Notes in Mathematics book series (LNM,volume 1394)

Abstract

We consider semilinear parabolic systems of partial differential equations of the form

$$u_t \left( {t,x} \right) = D\Delta u\left( {t,x} \right) + f\left( {u\left( {t,x} \right)} \right)t > 0,x \in \Omega$$
((1))

with bounded initial data and homogeneous Neumann boundary conditions, where D is an m by m diagonal positive definite matrix, Ω is a smooth bounded region in R n and f:R mR m is locally Lipschitz. We prove that if the vector field f satisfies a generalized Lyapunov type condition then either at least two components of the solution of (1) becomes unbounded in finite time or the solution exists for all t>0. Our result generalizes a recent result of Hollis, Martin, and Pierre [4], and the proof given is considerably simpler.

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References

  1. N. D. Alikakos, L p-bounds of solutions of reaction-diffusion equations, Comm. Partial Diff.Eq. 4, No. 8 (1979), 827–868.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. M. Feinberg, Complex balancing in general kinetic systems, Arch Rational Mech. Anal. 49 (1972), 187–194.

    CrossRef  MathSciNet  Google Scholar 

  3. K. Groger, On the existence of steady states of certain reaction-diffusion systems, Archive Rational Mechanics and Analysis, 1986.

    Google Scholar 

  4. S. Hollis, R. Martin, and M. Pierre, Global existence and boundedness in reaction-diffusion systems, (preprint).

    Google Scholar 

  5. F. Horn & R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal. 47 (1972), 81–116.

    CrossRef  MathSciNet  Google Scholar 

  6. F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal. 49 (1972), 172–186.

    CrossRef  MathSciNet  Google Scholar 

  7. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs Vol. 33, Amer. Math. Soc., Providence, Rhode Island, 1968.

    Google Scholar 

  8. J. H. Lightbourne and R. H. Martin, Relatively continuous nonlinear perturbations of analytic semigroups, Nonlinear Anal.-T.M.A., 1(1977), 277–292.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-diffusion equations, Hokkaido Math. J., XII (1982), 360–370.

    MathSciNet  MATH  Google Scholar 

  10. J. Morgan, Global existence for semilinear parabolic systems, (preprint).

    Google Scholar 

  11. R. Sperb, Maximum Principles and Their Applications, Mathematics in Science and Engineering, Vol. 157, Academic Press, New York, 1981.

    MATH  Google Scholar 

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© 1989 Springer-Verlag

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Morgan, J. (1989). Global existence for semilinear parabolic systems via Lyapunov type methods. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086756

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  • DOI: https://doi.org/10.1007/BFb0086756

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51594-4

  • Online ISBN: 978-3-540-46679-6

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