13A (Mathematical Biology)

Put

ξ = r / (r_{0} λ)

and

n = n_{0} λ^{- 2} F (ξ)

n_{t} = - n_{0} λ^{- 3} \dot{λ} (2 F + ξ F^{'}), \frac{D _{0}}{r} \partial_{r} [r (n / n_{0})^{2} n_{r}] = \frac{D _{0} n _{0}}{r _{0}^{2} λ ^{8}} \frac{1}{ξ} (ξ F^{2} F^{'})^{'} .

λ^{5} \dot{λ} = C D_{0} / r_{0}^{2}

. The profile then satisfies

(ξ F^{2} F^{'})^{'} = - C (ξ^{2} F)^{'}

. Regularity and zero radial flux at the origin make the integration constant zero. Wherever

F > 0

F F^{'} = - C ξ

, so

F^{2} = 1 - C ξ^{2}

using

F (0) = 1

. Requiring the front at

ξ = 1

selects

C = 1

. Point release requires

λ (0) = 0

λ (t) = (\frac{6 D _{0} t}{r _{0}^{2}})^{1/6}, F (ξ) = {1 - ξ^{2}, 0, 0 \leq ξ < 1, ξ \geq 1.

Although

F^{'}

diverges at the front,

F^{2} F^{'}

tends to zero, so this compactly supported profile satisfies the equation in the weak solution sense.

Q = 2 π \int_{0}^{\infty} r n (r, t) d r = 2 π n_{0} r_{0}^{2} \int_{0}^{1} ξ 1 - ξ^{2} d ξ = \frac{2 π n _{0} r _{0}^{2}}{3} .

Consequently $r_{0} = 3 Q / (2 π n_{0})$ . The front radius grows as

t^{1/6}

, while the central density decays as

t^{- 1/3}

. The conserved mass concentrates at the origin as

t ↓ 0

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