A semilinear parabolic system with a free boundary

Let be solution of equation (11) with ,
and . Then

Uniqueness of solution
First we get a new view for the free boundary. We rewrite the equation (1) as
(12)
Integrating the equation (12) over the domain $ D$, we find
(13)
In the same way, we rewrite the equation (2) as
(14)
Integrating the equation (14) over the domain D, we find
(15)

Now, multiplying (13) by (15) by and then adding them from the Stefan condition (5) we have

(16)
Theorem 5. Under the conditions of the Theorem 1, the solutions to the problem (1) – (8) are unique.
Proof. The theorem is proved first for small , and then it is established that it holds for any .
Let the functions and be solutions to the problem (1), (2) let it go . For each group of solutions, the representation (18).
Considering the difference, we find
$
|s_1(t)-s_2(t)| \leq \mu_1\int\limits_{-l}^{y(t)}|u_1(\xi,t)-u_2(\xi,t)| d\xi+\mu_2\int\limits_{y(t)}^{l}|v_1(\xi,t)-v_2(\xi,t)| d\xi + $$
$$+\mu_1\int\limits_{0}^{t} d\eta \int\limits_{-l}^{y(\eta)}|u_1(a_1-b_1u_1)-u_2(a_1-b_1u_2)| d\xi+ \mu_1\int\limits_{y(\eta)}^{h(\eta)}u_i(\xi,t)d\xi+ $$
$$+\mu_2 \int\limits_{0}^{t} d\eta \int\limits_{ y(\eta)}^{l}|v_1(a_2-b_2v_1)-v_2(a_2-b_2u_2)| d\xi+\mu_2\int\limits_{y(\eta)}^{h(\eta)}v_i(t,\xi)d\xi+$$
$$+\mu_1\int\limits_{0}^{t} d\eta \int\limits_{y(\eta)}^{h(\eta)}|u_i(a_i-b_iu_i)| d\xi+\mu_2\int\limits_{0}^{t} d\eta \int\limits_{y(\eta)}^{h(\eta)}|v_i(a_i-b_iv_i)| d\xi, \end{equation}
where $u_i,\quad v_i \,\,(i=1,2)$ -- decisions between $y(t)$ and $h(t)$, t.e
$$
({u_i\left( {x,t} \right),\quad v_i\left( {x,t} \right)}) = \left\{\begin{array}{rl}
({u_1\left( {x,t} \right),\quad v_1\left( {x,t} \right)}), & \textrm{if} \quad {s_2}\left( t \right) < {s_1}\left( t \right), \\
({u_2\left( {x,t} \right), \quad v_2\left( {x,t} \right)}),& \textrm{if} \quad {s_2}\left( t \right) > {s_1}\left( t \right).
\end{array} \right.
$$

By virtue of Theorem \ref{thm1}, we have

$$
\left| {{u_i}\left( {x,t} \right)} \right| \leq N_{1}\left( {y\left( t \right) - x} \right),
$$
$$
\left| {{v_i}\left( {x,t} \right)} \right| \leq N_{2}\left( { x -y\left( t \right)} \right),
$$
$$
\left| {{u_1}\left( {y\left( t \right),t} \right) - {u_2}\left( {y\left( t \right),t} \right)} \right| \leq M_4\left| {{s_1}\left( t \right) - {s_2}\left( t \right)} \right|,
$$
$$
\left| {{v_1}\left( {y\left( t \right),t} \right) - {v_2}\left( {y\left( t \right),t} \right)} \right| \leq M_5\left| {{s_1}\left( t \right) - {s_2}\left( t \right)} \right|,
$$
where $M_4=\mathop {\max }\limits_{D_{1}}|u_x(x,t)|$,\quad $M_5=\mathop {\max }\limits_{D_{2}}|v_x(x,t)|$.

For difference $V(x,t) = v_1(x,t)-v_2(x,t)$,\,\, $U(x,t) = u_1(x,t)-u_2(x,t)$ we get the following tasks

\begin{equation} \label{***2345}
\begin{cases}
U_{xx} -c_1U_x +A_1(x,t)U=U_t \,\,\, \text {in}\,\,\,D_1,\\
U(x,0) = 0,\, -l \leq x \leq 0, \\
U(-l, t) = 0,\, 0 \leq t \leq T,\\
U(y(t),t)\leq M_4 \mathop {\max }\limits_{0\leq \eta \leq t}|s_1(\eta)-s_2(\eta)|,
\end{cases}
\end{equation}
\begin{equation} \label{***23456}
\begin{cases}
V_{xx} -c_2V_x +A_1(x,t) V=V_t \,\,\, \text {in}\,\,\,D_2,\\
V (x,0) = 0,\, 0 \leq x \leq l, \\
V(l,t)=0,\, 0 \leq t \leq T,\\
V(y(t),t)\leq M_5 \mathop {\max }\limits_{0\leq \eta \leq t}|s_1(\eta)-s_2(\eta)|,
\end{cases}
\end{equation}
where are the coefficients $A_1(x,t)=a_1-b_1(u_1+u_2), \quad A_2(x,t)=a_2-b_2(v_1+v_2), $ -- limited and continuous functions.

From the problem \eqref{***2345} and \eqref{***23456} by the principle of maximum we find estimates

\begin{equation} \label{***23}
|U(x,t)| \leq N_{1} \max\limits_{0 \leq \eta \leq t}|s_1(\eta)-s_2(\eta)|,
\end{equation}
\begin{equation} \label{***24}
|V(x,t)| \leq N_{2} \max\limits_{0 \leq \eta \leq t}|s_1(\eta)-s_2(\eta)|. \end{equation}

By virtue of the established estimates for the functions $u\left( {x,t} \right)$, $v\left( {x,t} \right)$ $s\left( t \right)$ can rate members from \eqref{***22}.

We will use the results of\cite{17}. Since the single-phase case was investigated in \cite{17}, the integral terms in \eqref{***22} coincide with the corresponding terms.

Therefore, using the results of \cite{17} we complete the proof of the uniqueness theorem.

\section{ The existence solutions}

If you impose some restrictions (ensuring the fulfillment of the above factors at an arbitrary interval

time) for these problems, then the classical solution to the Stefan problem exists for all positive values
time.

\begin{thm} \label{thm6} Let the conditions of the theorems be satisfied Theorem \ref{thm1} and Theorem \ref{thm2} then there is a solution $u(x,t) \in C^{2+\gamma}\left(\overline{D}_{T}\right)$, $v(x,t) \in C^{2+\gamma}\left(\overline{Q}_{T}\right)$, $s(t) \in C^{1+\gamma}\left([0,T]\right)$ problem \eqref{1}-\eqref{8}.\end{thm}

\textbf{Proof.} Since the region under consideration does not degenerate, the Gelder derivative of $ \dot {s} (t) $ is also proved a priori estimates of the norms in the space $ C ^ {2+ \gamma} $ for $ v (t, x) $ are obtained, then we can prove the global solvability of the problem \cite{17}.

For this, we consider the equivalent problem \eqref{*13} to the problem \eqref{1} - \eqref{8}.

Since the coefficients of this equation satisfy the Holder condition \eqref{*13}, by virtue of the results on linear

we find the estimate
$$
| U | _ {2+ \alpha} ^ {\bar {Q}} \leq C.
$$

In the presence of the necessary a priori estimates for the free boundary and the solution of equations, methods of proof are developed.

global solvability of tasks. Since we have already established these estimates, by virtue of the results of \cite{8} statement of the theorem.

\begin{center}

\textbf{References}
\end{center}

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