On a Structure Theorem for Some Free Boundary Problems for the Heat Equation

In one space dimension and for a given initial datum https://www.w3.org/1998/Math/MathML"> u I ( x ) ∈ C 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755518/fbcb5fb2-fbc4-4b15-ab17-e703fd7c937e/content/eq660.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (say such that u_I(x) > 1 in some interval), the equation (1.1) https://www.w3.org/1998/Math/MathML"> u t = Δ ( u − 1 ) + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755518/fbcb5fb2-fbc4-4b15-ab17-e703fd7c937e/content/eq661.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>