## Free boundary problems for some reaction-diffusion equations.(English)Zbl 0649.35089

The problem is to find $$u(x,t)$$, $$s(t)$$, $$(x,t)\in [0,1]\times [0,\infty)$$ such that $u_ t=d_ 1u_{xx}+uf(u),\quad x<s(t);\quad u_ t=d_ 2u_{xx}+ug(u),\quad x>s(t);$
$u(0,t)=u(1,t)=0,\quad t>0;\quad u(s(t),t)=0,\quad t>0,$
$s(t)=-\mu_ 1u_ x(s(t)- 0,t)+\mu_ 2u_ x(s(t)+0,t)\text{ for } t>0 \text{ where } 0<s(t)<1,$
$u(x,0)=\phi (x),\quad 0<x<1;\quad s(0)=\ell.$ In this setting $$d_ i$$ and $$\mu_ i$$ are positive constants, and $$f,g,\phi$$, and $$\ell$$ are subject to the following conditions:
f is locally Lipschitz continuous on $$[0,\infty)$$, $$f(1)=0$$, $$f(u)\leq 0$$ on $$[1,\infty),$$
g is locally Lipschitz continuous on $$(-\infty,0]$$, $$g(-1)=0$$, $$g(u)\leq 0$$ on $$(-\infty,-1]$$
$$0\leq \ell \leq 1$$, $$\phi \in H^ 1_ 0(0,1)$$, $$\phi(\ell)=0$$, ($$\ell- x)\phi (x)\geq 0,0\leq x\leq 1.$$
This problem of the Stefan type with sources was studied by the authors in previous papers, under non-homogeneous Dirichlet data, where the free boundary s(t) never touches $$x=0,1$$. In the present case it is possible that one phase disappear in finite time.
The authors study global existence, uniqueness, regularity and asymptotic properties of solutions. Attention is given to the possibility that $$s(T^*)$$ be either 0 or 1 for certain $$T^*$$: the authors show that in that case s(t) will be identically 0 or 1 for $$t\geq T^*$$, and therefore the problem becomes an initial-boundary value problem for $$t\geq T^*$$. Comparison theorems are given, and results on the dependence of solutions on the initial data $$\phi$$,$$\ell$$ are proven. These results are used to study the $$\omega$$-limit set corresponding to the solution orbit of the problem: every element in this set satisfies the associated stationary problem. Stability and bifurcation are also studied.
Reviewer: J.E.Bouillet

### MSC:

 35R35 Free boundary problems for PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B32 Bifurcations in context of PDEs