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

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 → ∞.