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mth-Order (m > 1) Fisher Nonlinearity: Initial Data with Algebraic Decay Rates

  • J. A. Leach
  • D. J. Needham
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we extend the analysis of Chapter 3 by considering initial-boundary value problem [P,m] for m > 1, namely,
$$ \left. {\begin{array}{*{20}c} {u_t = u_{xx} + u^{_m } + (1 - u), x,t > (P1)} \\ {u(x,0) = u_0 (x), x \geqslant 0,(P2)} \\ {u_x (0,t) = 0, t > 0, (P3)} \\ {u(x,t) \to 0 as x \to \infty , t \geqslant 0, (P4)} \\ \end{array} } \right\}[P, m] $$
when the initial data u(x), is analytic, positive and monotone decreasing function in x ≥ 0, with albebraic decay (up to exponential corrections) of degree \( \left( { \geqslant \frac{1} {{m - 1}}} \right) \) as x → ∞ where
$${{u}_{0}}(x)\sim \left\{ \begin{gathered} \tfrac{{{{u}_{\infty }}}}{{{{x}^{\alpha }}}} + EST(x)asx \to \infty \left( {g1} \right) \hfill \\ {{{\tilde{u}}}_{0}} + \sum\nolimits_{{n = 1}}^{\infty } {{{{\tilde{u}}}_{n}}{{x}^{n}}asx \to {{0}^{ + }}(g2)} \hfill \\ \end{gathered} \right.$$
for some \( \left( { \geqslant \frac{1} {{m - 1}}} \right) \), where u , ũ 0 > 0 and ũ n are contants, and EST(x) denotes exponentially small terms in x as x → ∞.

Keywords

Asymptotic Solution Wave Speed Asymptotic Region Matched Asymptotic Expansion Rapid Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2004

Authors and Affiliations

  • J. A. Leach
    • 1
  • D. J. Needham
    • 1
  1. 1.Department of MathematicsThe University of ReadingReadingUK

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