In this paper we deal with the one-dimensional Stefan problemut−uxx
=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 ]Meγ[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 ]
Ceγ1[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 ]Meγ[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 ].
Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system
