Critical exponent in a Stefan problem with kinetic condition

Research Report MRR96-048

Critical exponent in a Stefan problem with kinetic condition

Zhicheng Guan Xu-Jia Wang

Abstract: In this paper we deal with the one-dimensional Stefan problem $$ u_t-u_{xx}= \dots (t) \delta(x-s (t)) \text{ in } \R\times\R^+ , u(x,0)=u_0(x)$$ with kinetic condition $\dots (t) =f(u)$ on the free boundary $F=\{(x,t), x=s(t) \}$, where $\delta (x)$ is the Dirac function. We proved in [10] that if $|f(u)|\le M\,e^{\gamma |u|}$ for some $M>0$ and $\gamma\in (0, 1/4)$, then there exists a global solution to the above problem; and the solution may blow up in finite time if $f(u)\ge Ce^{\gamma_1 |u|}$ for some $\gamma_1$ large. In this paper we obtain the optimal exponent, which turns out to be $\sqrt{2\pi e}$. That is, the above problem has a global solution if $|f(u)|\le M\,e^{\gamma |u|}$ for some $\gamma\in (0, \sqrt{2\pi e})$, and the solution may blow up in finite time if $f(u)\ge Ce^{\sqrt{2\pi e} |u|}$.

This service is maintained by the Mathematical Sciences Institute (MSI)
Comments to webmaster@maths.anu.edu.au URL: http://wwwmaths.anu.edu.au/