ABSTRACT
We study the singular Cauchy problem v t = ϕ ( v ) X X , ( x , t ) ∈ R × [ 0 , T ] , T > 0 v ( x , 0 ) = g ( x ) , x ∈ R . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072683/4382fcd0-4153-46e7-888a-e0b6b9eb509e/content/eq496.tif"/>
The constitutive function ψ(ξ) = max(0,ξ); the initial datum g is smooth (bounded) on R\{0} and satisfies g ( 0 ) = 0 ; x g ( x ) ≥ 0 , x ∈ R ; g ′ ( 0 + ) ⋅ g ′ ( 0 − ) ≠ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072683/4382fcd0-4153-46e7-888a-e0b6b9eb509e/content/eq497.tif"/>
where the superscript “+” (“-”) denotes the limit from the right (left). We show that the free boundary s, given by v(s(t)+,t) = 0, satisfies s ( t ) = − k t + o ( t ) ( t → 0 + ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072683/4382fcd0-4153-46e7-888a-e0b6b9eb509e/content/eq498.tif"/>
where κ > 0 is a monotone function of ρ = g′(0+)/g′(0−), implicitly defined by the equation p = κ 2 2 + κ 3 4 e k 2 / 4 ∫ − k ∞ e − t 2 / 4 d t , κ > 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072683/4382fcd0-4153-46e7-888a-e0b6b9eb509e/content/eq499.tif"/>
This generalizes an earlier analysis [7] for the special case of smooth data g in which g′(0+) = g′(0−) ≠ 0, and p = 1. In this case the numerical value κ =.903446… as computed from (*) is consistent with our previous result.
