Advertisement

The Existence for RDE with Small Delay

Conference paper
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 161)

Abstract

In this article, we prove that for the classical reaction-diffusion equations (RDE) u t – Δu = f(u(t), u(tτ)) with a small delay, there exists a weak solution. Our method is Galerkin approximations which is one of the most important methods in proving the existence of weak solution. This method can be found in many works, for example [4]. It is to build a weak solution by first constructing solutions of certain finite-dimensional approximations, and then building energy estimate and last passing to limits. Imposing some condition on the nonlinear f, we first make use of Galerkin approximations to prove the local existence and uniqueness theorem for weak solutions of the Initial-Bounded Value Problem.

Keywords

reaction-diffusion equations with small delay Galerkin approximations energy estimates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hale, J.: Asymptotic Behaviour of dissipative systems. Math. Surv. Monogr. AMS 20 (1988)Google Scholar
  2. 2.
    Henry, D.: Geometric Theory of Semi-linear Parabolic Equations. Springer Lecture Notes in Mathematics, vol. 840 (1981)Google Scholar
  3. 3.
    Matano, H.: Asymptotic behaviour and stability of solutions of semi-linear diffusion equations. Publ. Res. Inst. Math. Sci. 15, 401–458 (1979)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19Google Scholar
  5. 5.
    Friesecke, C.R.: Convergence to equilibrium for delay-diffusion equations with small delay. J. Dynam. Differential Equations 5, 89–103 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.College of ScienceTianjin Polytechnic UniversityTianjinChina

Personalised recommendations