Consider one-phase Stefan problem 1 with unknowns \(u(t,x)\) and free boundary \(h(t)\) \[ u_t - d u_{xx} = u(a-bu), \quad x\in (0,h(t)),\;t>0, \]
\[ h(0)=h_0, \quad u(0,x)= u_0(x), \;x\in [0,h_0], \]
\[ u_x(t,0)=0,\;u(t,h(t))=0,\quad h'(t)=-\mu u_x(t,h(t)), \;t>0, \] where \(d, a, b, h_0, \mu\) are positive constants.
The authors prove that for all \(t\in(0,\infty)\) this problem has a unique classical solution, such that \(0\leq h'(t)\leq C\), \(C\)=const \(>0\). The asymptotic behavior of the solution \(u, \;h\) as \(t\to \infty\) is studied. For the problem of Stefan type with two free boundaries \(g(t), h(t)\), \(g(0)=-h_0, \;h(0)=h_0\), in the domain \((g(t), h(t)), \;t>0\) the similar results as for the problem 1 are obtained, in particular, it is shown that \(g'(t)<0\), \(h'(t)>0\) for \(t>0\).