Critical exponent in a Stefan problem with kinetic condition

Abstract

In this paper we deal with the one-dimensional Stefan problem

u_t−u_xx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ⁺, u(x, 0) =u₀(x)

with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Me^{γ[mid ]u[mid ]} for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ce^{γ₁[mid ]u[mid ]} for some γ₁ large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Me^{γ[mid ]u[mid ]} for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce^{√2πe[mid ]u[mid ]}.