Wavefronts for convection-diffusion

Abstract

As mentioned earlier, when the reaction term in (1.1) is absent and the partial differential equation is

$$ {u_t} = {\left( {a\left( u \right)} \right)_{xx}} + {\left( {b\left( u \right)} \right)_x} $$

(9.1)

the integral equation (1.9) reduces to the simple identity θ(s) = σs+b(s) . Sub­sequently, if this ‘equation’ is to have a nonnegative solution on [0,ℓ] with ℓ < ∞ such that θ(ℓ)= 0, then necessarily σ = -b (ℓ) / ℓ and θ (s)= b (s) — sb (ℓ) / ℓ ≥ 0 for all 0 ≤ s ≤ℓ . By Lemma 2.40 though θ satisfies the integrability condition if and only if it is positive on (0, ℓ). We conclude the following.