UDC 517.945.43

MATHEMATICS

B. M. BUDAK, M. Z. MOSKAL

ON THE CLASSICAL SOLUTION OF A MULTIDIMENSIONAL MULTIFRONT STEFAN PROBLEM

(Presented by Academician A. N. Tikhonov, February 5, 1969)

In the present note we prove the existence, uniqueness, and stability with respect to perturbations of the initial data of a regular classical solution of a multidimensional multifront Stefan problem (Problem A) with internal and external phase fronts. The exposition is given for the case when the number of spatial independent variables is \(N=2\), but the method and the results are easily extended to the case \(N>2\).

1°. Problem A. Find the functions \(u^i=u^i(x_1,x_2,t)\), \(x_1=x_1^i(s,t)\), \(x_2=x_2^i(s,t)\), \(0\le s\le s_0\), \(0\le t\le T\), \(i=1,2,\ldots,n\), from the conditions

\[ u_t^i=a_i^2\left(u_{x_1x_1}^i+u_{x_2x_2}^i\right)+F^i(x_1,x_2,t) \quad \text{for } (x_1,x_2,t)\in D_T^i,\quad i=1,\ldots,n, \tag{1} \]

where \(\overline D_T^i\) is a closed simply connected domain of values of \((x_1,x_2,t)\), whose boundary consists of four simply connected plane pieces \(D_{0x_1x_2}^i\), \(D_{Tx_1x_2}^i\), \(D_{tx_1 0}^i\), \(D_{tx_1 l_{22}}^i\), lying respectively in the planes \(t\equiv 0\), \(t\equiv T\), \(x_2\equiv l_{12}=0\), \(x_2\equiv l_{22}\), and two curved simply connected pieces
\[ S_T^j=\{x_1=x_1^j(s,t),\; x_2=x_2^j(s,t),\; 0\le s\le s_0,\; 0\le t\le T\}, \]
\(j=i,i+1\); \(i=1,2,\ldots,n\), with \(S_T^k\), \(k=1,\ldots,n\), being the unknown surfaces (phase fronts), and \(S_T^{n+1}\) a given surface; \(D_T^i=\overline D_T^i\setminus \Gamma_T^i\),

\[ \Gamma_T^i=D_{0x_1x_2}^i\cup D_{tx_1 0}^i\cup D_{tx_1 l_{22}}^i\cup S_T^i\cup S_T^{i+1}; \]

\[ u^i(x_1,x_2,0)=\varphi_i(x_1,x_2),\quad (x_1,x_2)\in D_{0x_1x_2}^i,\quad i=1,2,\ldots,n; \tag{2} \]

\[ u^k\big|_{S_T^i} = u^k(x_1,x_2,t)\big|_{x_1=x_1^i(s,t),\,x_2=x_2^i(s,t)} = \omega_k^i(s,t) = \omega_k^i\bigl(x_1^i(s,t),x_2^i(s,t),t\bigr); \]

\[ i=k,k+1;\quad k=1,\ldots,n;\qquad \omega_{k-1}^k\equiv \omega_k^k,\quad k=2,\ldots,n-1. \tag{3′} \]

\[ u^k\big|_{D_{tx_1 l_{j2}}^k} = u^k(x_1,x_2,t)\big|_{x_2=l_{j2}} = f_k^j(x_1,t),\quad j=1,2; \]

\[ f_k^j(x,t)\ \text{is defined in }D_{tx_1 l_{j2}}^k;\quad j=1,2;\ k=1,\ldots,n; \tag{3″} \]

\[ -\frac{\partial x_i^j(s,t)}{\partial t} = \overline{\lambda}^{\,jj-1}(s,t) \frac{\partial u^{j-1}\bigl(x_1^j(s,t),x_2^j(s,t),t\bigr)}{\partial x_i} - \overline{\lambda}^{\,jj}(s,t) \frac{\partial u^j\bigl(x_1^j(s,t),x_2^j(s,t),t\bigr)}{\partial x_i}, \tag{4} \]

\[ i=1,2;\quad j=1,2,\ldots,n,\qquad \text{where } u_{x_1}^0\equiv 1,\quad u_{x_2}^0\equiv 1; \]

\[ \overline{\lambda}^{\,jk} = \lambda^j\bigl(s,t,x_1^k(s,t),x_2^k(s,t)\bigr), \quad k=j-1,j;\ j=1,2,\ldots,n; \tag{5} \]

\[ x_i^j(s,0)=\psi_i^j(s),\quad 0\le s\le s_0,\quad i=1,2;\ j=1,2,\ldots,n+1. \]

Under the assumption that the initial curves
\[ \Gamma_j=\{x_1=\psi_1^j(s),\ x_2=\psi_2^j(s)\},\quad j=1,2,\ldots,n+1, \]
have no common points, we shall seek a solution of Problem A \(((1)-(5))\) on such a time interval \(0\le t\le T^*\) on which \(S_{T^*}^i\), \(i=1,\ldots,n+1\), have no common points and the condition

\[ \bigl[x_{1s}^i(s,t)\bigr]^2+\bigl[x_{2s}^i(s,t)\bigr]^2\ne 0 \tag{6} \]

is satisfied.

Definition 1. The surface \(S_T^i \equiv \{x_1=x_1^i(s,t),\ x_2=x_2^i(s,t),\ 0\le s\le s_0,\ 0\le t\le T\}\) is said to satisfy requirements B if:

1) \(S_T^i\) is a simple unclosed smooth surface; in particular, \(x_{1t}^i(s,t)\), \(x_{2t}^i(s,t)\), \(x_{1s}^i(s,t)\), \(x_{2s}^i(s,t)\) are continuous and
\[ [x_{1s}^i(s,t)]^2+[x_{2s}^i(s,t)]^2\ne 0 \]
for \(0\le s\le s^0,\ 0\le t\le T\), and all interior points of \(S_T^i\) are interior points of the cylinder
\[ C\equiv \{0\le x_2\le l_{22},\ 0\le t\le T,\ -\infty<x_1<+\infty\}; \]

2) the boundary of the surface \(S_T^i\) is a simple continuous, closed piecewise-smooth curve consisting of four smooth pieces lying respectively on the faces \(x_2=0\), \(x_2=l_{22}\), \(t=0\), \(t=T\) of the cylinder \(C\).

Definition 2. A regular classical solution of problem A is a system of functions
\[ u^i=u^i(x_1,x_2,t),\quad x_1=x_1^i(s,t),\quad x_2=x_2^i(s,t),\quad 0\le s\le s_0,\quad 0\le t\le T,\quad i=1,2,\ldots,n, \]
satisfying the conditions: 1) the surfaces \(S_T^i\) satisfy requirements B and have no common points with one another or with \(S_T^{n+1}\); 2) \(u^i=u^i(x_1,x_2,t)\) are continuous in \(\overline{D_T^i}\), \(i=1,2,\ldots,n\); \(u_{x_1}^i(x_1,x_2,t)\), \(u_{x_2}^i(x_1,x_2,t)\), \(u_{x_1x_1}^i(x_1,x_2,t)\), \(u_{x_1x_2}^i(x_1,x_2,t)\), \(u_{x_2x_2}^i(x_1,x_2,t)\), \(u_t^i(x_1,x_2,t)\) are continuous in \(D_T^i\), \(i=1,2,\ldots,n\), and take uniformly continuous limiting values at the interior points of the surfaces \(S_T^j\), \(j=i,\ i+1\); 3) all relations (1)—(5) of problem A and condition (6) are satisfied.

Remark. If \((u^i,x_1^i,x_2^i,\ i=1,\ldots,n)\) is a regular classical solution of problem A, then it is also a classical solution of problem A in the following sense: on each phase front \(S_T^i\), represented by the equation \(\Phi^i(x_1,x_2,t)=0\) (which is possible in a neighborhood of each point \(\in S_T^i\), since by the condition
\[ [x_{1s}^i(s,t)]^2+[x_{2s}^i(s,t)]^2\ne 0 \]
for \(0\le s\le s_0,\ 0\le t\le T,\ i=1,2,\ldots,n\)), the relation
\[ \Phi_t^i+\bigl(\lambda^{ii-1}\operatorname{grad}u^{i-1}-\lambda^{ii}\operatorname{grad}u^i,\operatorname{grad}\Phi^i\bigr)=0,\quad i=1,2,\ldots,n. \]

\(2^0.\) We reduce problem A to a system of nonlinear integral equations of the second kind. To this end consider the functions
\[ G_{1k}(x_1,t,\xi_1,\tau)= \frac{1}{2a_k\sqrt{\pi(t-\tau)}}\exp\left[-\frac{(x_1-\xi_1)^2}{4a_k^2(t-\tau)}\right], \quad k=1,2,\ldots,n; \]
\[ G_{2k}(x_2,t,\xi_2,\tau)= \frac{1}{2a_k\sqrt{\pi(t-\tau)}} \sum_{n=-\infty}^{+\infty} \left\{ \exp\left[-\frac{(x_2-\xi_2+2nl_{22})^2}{4a_k^2(t-\tau)}\right] -\exp\left[-\frac{(x_2+\xi_2-2nl_{22})^2}{4a_k^2(t-\tau)}\right] \right\}, \quad k=1,\ldots,n; \]
\[ \Pi_k(x_1,x_2,t,\xi_1,\xi_2,\tau) =G_{1k}(x_1,t,\xi_1,\tau)G_{2k}(x_2,t,\xi_2,\tau), \quad k=1,\ldots,n; \tag{7} \]
\[ v_m^{kl}(s,t)=u_{x_m}^k(x_1^l(s,t),x_2^l(s,t),t),\quad g_{mr}^{kl}(s,t)=u_{x_mx_r}^k(x_1^l(s,t),x_2^l(s,t),t), \tag{8} \]
\[ m,r=1,2;\quad l=k,k+1;\quad k=1,2,\ldots,n. \]

Integrating Green’s identity for \(\Pi_k\) and \(u^k\) over \(D_t^k\) from (1), (2), and (3), we obtain, for the regular classical solution \(u^k\) of problem A, the representation
\[ u^k(x_1,x_2,t)= \iint_{D_{0x_1x_2}^k} \varphi_k(\xi_1,\xi_2)\Pi_k\,d\xi_1d\xi_2 -a_k^2\sum_{i=1}^{2} \iint_{D_{tx_1^i l_2^i}^{k}} f_k^i(\xi_1,\tau)\Pi_{k\xi_2}\,d\xi_1d\tau + \]
\[ +\sum_{i=k}^{k+1} \iint_{S_t^i} \bar{\omega}_k^i(s,\tau)\Pi_k\,d\xi_1d\xi_2 -a_k^2\sum_{j=1}^{2}\sum_{i=k}^{k+1} \iint_{S_t^i} \bar{\omega}_k^i(s,\tau)\Pi_{k\xi_j}\,d\xi_{\rho(j)}d\tau + \]

\[ + a_k^2 \sum_{j=1}^{2} \sum_{i=k}^{k+1} \iint_{S_t^i} v_j^{ki}(s,\tau)\Pi_k\,d\xi_{p(j)}d\tau + \iiint_{D_t^k} F_k(\xi_1,\xi_2,\tau)\Pi_k\,d\xi_1d\xi_2d\tau, \]

\[ p(j)=2-2^{-1}[1+(-1)^j],\qquad k=1,\ldots,n. \tag{9} \]

Next, consider the system (10)—(12) of nonlinear Volterra integral equations of the second kind for the functions \(v_m^{kl}(s,t)\), \(x_i^k(s,t)\), \(x_{is}^k(s,t)\), \(i=1,2;\ l=k,k+1;\ k=1,2,\ldots,n\)

\[ v_m^{kl}= \iint_{D_{0x_1x_2}^k}\varphi_k\Pi_{kx_m}\,d\xi_1d\xi_2 -a_k^2\sum_{i=1}^{2}\iint_{D_{tx_1l_i^2}^k} f_k^i\Pi_{k\xi_2x_m}\,d\xi_1d\tau+ \]

\[ +\sum_{i=k}^{k+1}\iint_{S_t^i}\overline{\omega}_k^i\Pi_{kx_m}\,d\xi_1d\xi_2 -a_k^2\sum_{j=1}^{2}\sum_{i=k}^{k+1}\iint_{S_t^i}\overline{\omega}_k^i\Pi_{k\xi_jx_m}\,d\xi_{p(j)}d\tau+ \]

\[ +\sum_{j=1}^{2}\sum_{i=k}^{k+1}\iint_{S_t^i}v_j^{ki}\Pi_{kx_m}\,d\xi_{p(j)}d\tau +\iiint_{D_t^k}F_k\Pi_{kx_m}\,d\xi_1d\xi_2d\tau; \tag{10} \]

\[ p(j)=2-2^{-1}[1+(-1)^j],\qquad m=1,2;\quad l=k,k+1;\quad k=1,\ldots,n; \]

\[ x_i^j(s,t)=\psi_i^j(s)+\int_0^t \left[\overline{\lambda}^{\,jj-1}(s,\tau)v_i^{jj-1}(s,\tau) -\overline{\lambda}^{\,jj}(s,\tau)v_i^{jj}(s,\tau)\right]\,d\tau, \]

\[ j=1,\ldots,n;\quad i=1,2; \tag{11} \]

\[ x_{is}^j(s,t)=\psi_{is}^j(s)+\int_0^t \Bigg\{\overline{\lambda}_s^{\,jj-1}(s,\tau)v_i^{jj-1}(s,\tau) -\overline{\lambda}_s^{\,jj}(s,\tau)v_i^{jj}(s,\tau)+ \]

\[ +\sum_{k=j-1}^{j}(-1)^{j-k+1}\overline{\lambda}^{\,jk}(s,\tau) \left[g_{i1}^{kj}(s,\tau)x_{1s}^j(s,\tau)+g_{i2}^{kj}(s,\tau)x_{2s}^j(s,\tau)\right]\Bigg\}\,d\tau, \]

\[ i=1,2;\quad j=1,\ldots,n; \tag{12} \]

\[ \lambda_s^{jk}(s,t)=\frac{d}{ds}\lambda^j(s,t,x_1^k(s,t),x_2^k(s,t)); \tag{13} \]

\[ g_{mr}^{kl}(s,t)= \iint_{D_{0x_1x_2}^k}\varphi_k\Pi_{k\xi_2x_mx_r}\,d\xi_1d\xi_2 -a_k^2\sum_{i=1}^{2}\iint_{D_{tx_1l_i^2}^k}f_k^i\Pi_{k\xi_1x_mx_r}\,d\xi_1d\tau+ \]

\[ +\sum_{i=k}^{k+1}\iint_{S_t^i}\overline{\omega}_k^i\Pi_{kx_mx_r}\,d\xi_1d\xi_2 -a_k\sum_{j=1}^{2}\sum_{i=k}^{k+1}\iint_{S_t^i}\overline{\omega}_k^i\Pi_{k\xi_jx_mx_r}\,d\xi_{p(j)}d\tau- \]

\[ -a_k^2\sum_{j=1}^{2}\sum_{i=k}^{k+1}\iint_{S_t^i}v_j^{ki}\Pi_{kx_mx_r}\,d\xi_{p(j)}d\tau +\iiint_{D_t^k}F_k\Pi_{kx_mx_r}\,d\xi_1d\xi_2d\tau, \tag{14} \]

\[ m,r=1,2;\quad l=k,k+1;\quad p(j)=2-2^{-1}[1+(-1)^j];\quad k=1,2,\ldots,n; \]

\[ g_{mr}^{kl}(s,t)=g_{rm}^{kl}(s,t). \]

Definition 3. Problem A (1)—(5) is equivalent to system B (10)—(12) if every regular classical solution of problem A (1)—(5) generates, by virtue of (8) and (9), a continuous solution of system B (10)—(12), and conversely every continuous solution of system B (10)—(12), satisfying inequality (6), generates, by virtue of (8) and (9), a regular classical solution of problem A (1)—(5).

Theorem 1 (equivalence). Suppose that for the initial data of problem A the following conditions are satisfied:

1) \(F^k(x_1,x_2,t)\), \(F_{x_i}^k(x_1,x_2,t)\), \(F_t^k(x_1,x_2,t)\) are continuous with respect to \(x_1,x_2,t\), \(k=1,2,\ldots,n\);

2) \(\varphi_k(x_1,x_2)\), \(\varphi_{kx_i}(x_1,x_2)\), \(\varphi_{kx_ix_j}(x_1,x_2)\), \(k=1,2,\ldots,n;\ i,j=1,2,\) are continuous in \(x_1,x_2\);

3) \(f_k^i(x_1,t)\), \(f_{kx_1}^i(x_1,t)\), \(f_{kt}^i(x_1,t)\), \(f_{kx_1t}^i(x_1,t)\), \(i=1,2;\ k=1,2,\ldots,n,\) are continuous in \(x_1,t\);

4) \(\bar{\lambda}^{kh}(s,t)\), \(\bar{\lambda}^{kh-1}(s,t)\), \(\bar{\omega}^{\,ki}(s,t)\) are continuously differentiable with respect to all arguments;

5) \(\psi_i^j(s)\), \(\psi_{is}^j(s)\), \(j=1,\ldots,n;\ i=1,2,\) are continuous in \(s\); \([\psi_{1s}^j(s)]^2+[\psi_{2s}^j(s)]^2=0,\ j=1,\ldots,n+1;\ 0\le s\le s_0\);

6) \(x_i^{n+1}(s,t)\), \(x_{is}^{n+1}(s,t)\), \(x_{it}^{n+1}(s,t)\), \(i=1,2,\) are continuous; \([x_{1s}^{n+1}(s,t)]^2+[x_{2s}^{n+1}(s,t)]^2\ne0,\ 0\le s\le s_0,\ 0\le t\le T,\ i=1,2\).

7) The compatibility conditions for the initial and boundary data are satisfied.

Then problem A (1)—(6) is equivalent to the system of integral equations B (10)—(12), by virtue of relations (13), (14).

Theorem 2. Under the conditions of Theorem 1 there exists an interval of time \(0\le t\le T^*,\ 0<T^*\le T,\) on which there exists a unique regular solution of system B (10)—(12), and hence there exists a unique regular classical solution of problem A (1)—(5).

The proof of Theorem 2 is obtained by the method of contractions.

Definition 4. Suppose that, along with problem A (1)—(5), there is given problem \(A^*\) \((1^*)\)—\((5^*)\), differing from problem A (1)—(5) only in that all the initial data \(\varphi_k(x_1,x_2)\), \(f_k^i(x_1,t)\), \(F_k(x_1,x_2,t)\), \(a_k\), \(\psi_i^k(s)\), \(\bar{\lambda}^{kk-1}(x_1,x_2,s,t)\), \(\bar{\lambda}^{kk}(x_1,x_2,s,t)\), \(\bar{\omega}^{\,ki}(s,t)\) are replaced, respectively, by \(\varphi_k^*(x_1,x_2)\), \(f_k^{i*}(x_1,t)\), \(F_k^*(x_1,x_2,t)\), \(a_k^*\), \(\psi_i^{k*}(s)\), \(\bar{\lambda}^{kk-1*}(x_1,x_2,s,t)\), \(\bar{\lambda}^{kk*}(x_1,x_2,s,t)\), \(\bar{\omega}^{\,ki*}(s,t)\). Put

\[ \eta=\max\left\{ \max_k |a_k-a_k^*|,\ \max_{s,i,k}|\psi_1^k(s)-\psi_i^{k*}(s)|,\ \max_{s,i,k}|\psi_{is}^k(s)-\psi_{is}^{k*}(s)|,\right. \]

\[ \max_{x_1,x_2,s,t,k}\left|\bar{\lambda}^{kk-1}-\bar{\lambda}^{kk-1*}\right|,\ \max_{x_1,x_2,s,t,k}\left|\bar{\lambda}^{kk}-\bar{\lambda}^{kk*}\right|, \]

\[ \max_{x_1,x_2,k}|\varphi_k-\varphi_k^*|,\ \max_{x_1,x_2,i,k}|\varphi_{kx_i}-\varphi_{kx_i}^*|,\ \max_{x_1,x_2,i,j,k}|\varphi_{kx_ix_j}-\varphi_{kx_ix_j}^*|, \]

\[ \max_{x_1,i,k,t}|f_k^i(x_1,t)-f_k^{i*}(x_1,t)|,\ \max_{t,x_1,i,k}|f_{kx_1}^i(x_1,t)-f_{kx_1}^{i*}(x_1,t)|, \]

\[ \max_{t,x_1,i,k}|f_{kt}^i(x_1,t)-f_{kt}^{i*}(x_1,t)|,\ \max_{t,x_1,i,k}|f_{kx_1t}^i(x_1,t)-f_{kx_1t}^{i*}(x_1,t)|, \]

\[ \left. \max_{x_1,x_2,t,k}|F_k-F_k^*|,\ \max_{x_1,x_2,t,i,k}|F_{kx_i}-F_{kx_i}^*|,\ \max_{x_1,x_2,t}|F_{kt}-F_{kt}^*| \right\}, \]

\[ 0\le t\le T^*,\qquad 0<T^*\le T, \]

\[ \varepsilon=\max\left[ \max_{i,k,s,\,0\le t\le T^*}|x_i^k(s,t)-x_i^{k*}(s,t)|,\ \max_{s,i,k,\,0\le t\le T^*}|x_{is}^k(s,t)-x_{is}^{k*}(s,t)|,\right. \]

\[ \left. \max_{x_1,x_2,k,\,0\le t\le T^*}|u^k(x_1,x_2,t)-u^{k*}(x_1,x_2,t)| \right]. \]

The regular classical solution \(u^k(x_1,x_2,t)\), \(x_1^k(s,t)\), \(x_2^k(s,t)\), \(k=1,2,\ldots,n,\ 0\le s\le s_0,\ 0\le t\le T^*,\ 0<T^*\le T,\) of problem A (1)—(5) is called stable with respect to perturbations of the initial data if \(\varepsilon\to0\) as \(\eta\to0\).

Theorem 3. If the initial data of problems A and \(A^*\) satisfy the conditions of Theorem 1, then the regular classical solution of problem A is stable with respect to perturbations of the initial data.

Remark. All the results extend also to a quasilinear parabolic equation with principal part

\[ u_t^k=a_k^2\left(u_{x_1x_1}^k+u_{x_2x_2}^k\right) \]

and with a free term depending on \(x_1,x_2,t,u^k,u_{x_1}^k,u_{x_2}^k\), to quasilinear boundary conditions, and also to the case where, along with unclosed phase fronts, there are closed phase fronts.

Moscow State University
named after M. V. Lomonosov

Received
17 I 1969