F. P. Vasil'ev

A Difference Method for Solving Stefan-Type Problems for a Quasilinear Parabolic Equation with Discontinuous Coefficients

(Presented by Academician A. A. Dorodnitsyn on 10 February 1964)

In the present note we consider two-phase and one-phase Stefan-type problems for a quasilinear equation of parabolic type with discontinuous coefficients under nonlinear boundary conditions. These problems include spatial Stefan-type problems with radial symmetry in the cylindrical and spherical cases. In writing the present note, the method of the papers (1–6) has been used and generalized.

1°. Consider the following two-phase Stefan-type problem: find the functions $\bar u(x,t)$, $u(x,t)$, $y(t)$ satisfying the conditions:

\[ (\bar k(x)\bar u_x)_x-\bar\rho(x,t,\bar u)\bar u_t=0,\qquad 0<x<y(t),\quad x\ne l_m,\quad t>0; \tag{1} \]

\[ [\bar u(x,t)]_{x=l_m}=0,\qquad [\bar k(x)\bar u_x(x,t)]_{x=l_m}=0,\qquad t>0; \tag{2} \]

\[ \bar k(0)\bar u_x(0,t)=-\bar q(t,\bar u(0,t)),\qquad \bar u(y(t),t)=0,\qquad t>0; \tag{3} \]

\[ \bar u(x,0)=\bar\varphi(x),\qquad 0\le x\le c=y(0); \tag{4} \]

\[ (k(x)u_x)_x-\rho(x,t,u)u_t=0,\qquad y(t)<x<l,\quad x\ne l_m,\quad t>0; \tag{5} \]

\[ [u(x,t)]_{x=l_m}=0,\qquad [k(x)u_x(x,t)]_{x=l_m}=0,\qquad t>0; \tag{6} \]

\[ k(l)u_x(l,t)=-q(t,u(l,t)),\qquad u(y(t),t)=0,\qquad t>0; \tag{7} \]

\[ u(x,0)=\varphi(x),\qquad c\le x\le l; \tag{8} \]

\[ \gamma(y(t),t)y'(t)=k(y(t))u_x(y(t),t)-\bar k(y(t))\bar u_x(y(t),t)+\Phi(y(t),t) \tag{9} \]

for $t>0;\ y(0)=c$,

where $\bar k(x)$ and $k(x)\in Q$ $(x\ne l_m)$ for $0\le x\le l$; $\bar\rho(x,t,s)$ and $\rho(x,t,s)\in Q$ $(x\ne l_m)$ in
$G=\{0\le x\le l,\ t\ge0,\ |s|<\infty\}^{*}$; $[z(x,t)]_{x=l_m}=$
\[ = z(l_m+0,t)-z(l_m-0,t),\quad m=1,2,\ldots,m_1-1;\quad 0=l_0\le l_1\le\cdots \]
\[ \le\cdots l_{m_0-1}\le c<l_{m_0}\le l_{m_0+1}\le\cdots\le l_{m_1}=l,\quad m_1\ge m_0\ge1 \]

(the case $y(0)=0$ is possible); $\bar q(t,\bar u)$, $q(t,u)$, $\gamma(x,t)$, $\Phi(x,t)$, $\bar\varphi(x)$, $\varphi(x)$ are given functions; $\bar\varphi(c)=\varphi(c)=0$. We note that the two-phase Stefan-type problem for more general equations of the form

\[ (\bar\nu(r)\bar\lambda(\bar v)\bar v_r)_r-\bar c(r,t,\bar v)\bar v_t=0,\qquad r_0<r<R(t),\quad r\ne r_m,\quad t>0; \]

\[ (\nu(r)\lambda(v)v_r)_r-c(r,t,v)v_t=0,\qquad R(t)<r<r_m,\quad r\ne r_m,\quad t>0, \]

* We shall write $f(x,z_1,\ldots,z_n)\in C^{(s_0,s_1,\ldots,s_n)}$ in some domain $G$ of the variables $(x,z_1,\ldots,z_n)$ if all derivatives of the form
\[ \partial^{p_0+p_1+\cdots+p_n}f/\partial x^{p_0}\partial z_1^{p_1}\cdots\partial z_n^{p_n}, \quad 0\le p_i\le s_i,\quad 0\le p_0+p_1+\cdots+p_n\le \max\{s_0,s_1,\ldots,s_n\} \]
exist and are continuous jointly in their arguments everywhere in $G$. If, however, the indicated derivatives are continuous jointly in their arguments everywhere in $G$ except, possibly, at points of the form $(x=l_m,z_1,\ldots,z_n)\in G$ $(m=1,2,\ldots,m_1-1)$, where these derivatives may have jump discontinuities, then we shall write
\[ f(x,z_1,z_2,\ldots,z_n)\in Q^{(s_0,s_1,\ldots,s_n)}\quad (x\ne l_m). \]
In the case $s_0=s_1=\cdots=s_n=0$ we denote
\[ C^{(0,0,\ldots,0)}=C,\qquad Q^{(0,0,\ldots,0)}=Q. \]

where \(\bar v(r), v(r), \bar c(r,t,\bar v), c(r,t,v)\in Q(r\ne r_m), r_0>0\), is reduced to problem (1)—(9) by means of the substitution
\[ x=r-r_0,\quad y(t)=R(t)-r_0,\quad \bar u(x,t)=\int_0^{\bar v(r,t)} \bar\lambda(s)\,ds,\quad u(x,t)=\int_0^{v(r,t)} \lambda(s)\,ds,\quad \bar k(x)=\bar v(r),\quad k(x)=v(r). \]

Definition. We shall call the functions \(\bar u(x,t)\), \(u(x,t)\), \(y(t)\) a solution of problem (1)—(9) if: 1) \(y(t)\) is defined and continuous on some interval \(0\le t\le T\) and has a continuous derivative \(y'(t)\) for \(0<t\le T\); 2) \(\bar u(x,t)\) is defined and continuous in \(\{0\le x\le y(t),\ 0\le t\le T\}\); the derivative \(\bar u_x(x,t)\in Q(x\ne l_m)\) and is uniformly bounded in \(\{0\le x\le y(t),\ 0<t\le T\}\); the derivatives \(\bar u_{xx}(x,t)\), \(\bar u_t(x,t)\) are continuous and uniformly bounded in \(\{0<x<y(t),\ x\ne l_m,\ 0<t\le T\}\) \((m=1,2,\ldots,m_1-1)\); the function \(u(x,t)\) has analogous properties in \(\{y(t)\le x\le l,\ 0\le t\le T\}\); 3) all conditions (1)—(9) are satisfied.

For an approximate solution of problem (1)—(9), assuming \(y(t)\) monotone, we introduce a rectangular difference grid: we divide the interval \(0\le x\le l\) by points \(x_i\) into \(N\) equal parts with step \(h\), where \(x_i=ih,\ i=0,1,\ldots,N;\ x_N=l;\ x_{i_m}=i_mh=l_m\ (m=1,2,\ldots,m_1-1);\ c=i_ch\), and the time step \(\tau_n\) will be chosen so that for each
\[ t=t_n=\sum_{k=1}^{n}\tau_k\quad (t_0=0) \]
the end of a segment of the polygonal line approximating the curve \(y(t)\) falls on the node with coordinates \((y_n,t_n)\), \(y_n=c+nh,\ n=0,1,\ldots\). We replace problem (1)—(9) by the following difference problem for determining \(\tau_n\) and the approximate values \(\bar w_{in}, w_{in}\) of the functions \(\bar u(x,t), u(x,t)\) at the nodes \((x_i,t_n)\):
\[ \delta_{\bar x}\left(\bar k_i\delta_x\bar w_{in}\right)-\bar\rho_{i,n-1}\delta_t^-\bar w_{in}=0,\quad 1\le i\le i_c+n-1,\quad i\ne i_m \tag{10} \]
\[ (m=1,2,\ldots,m_0-1),\quad n=1,2,\ldots; \]
\[ \delta_{\bar x}\left(\bar k_{i_m}\delta_x\bar w_{i_m,n}\right)=0\quad (m=1,2,\ldots,m_0-1),\quad n=1,2,\ldots; \tag{11} \]
\[ \bar k_0\delta_x\bar w_{0,n}=-\bar q_{n-1},\quad \bar w_{i_c+n,n}=0,\quad n=1,2,\ldots; \tag{12} \]
\[ \bar w_{i0}=\bar\varphi_i=\bar\varphi(x_i),\quad 0\le i\le i_c; \tag{13} \]
\[ \delta_{\bar x}\left(k_i\delta_x w_{in}\right)-\rho_{i,n-1}\delta_t^- w_{in}=0,\quad i_c+n+1\le i\le N-1; \tag{14} \]
\[ i\ne i_m\ (m=m_0,m_0+1,\ldots,m_1-1),\quad n=1,2,\ldots; \]
\[ \delta_{\bar x}\left(k_{i_m}\delta_x w_{i_m,n}\right)=0\quad (m=m_0,\ldots,m_1-1),\quad n=1,2,\ldots; \tag{15} \]
\[ k_{N-1}\delta_x w_{N-1,n}=-q_{n-1},\quad w_{i_c+n,n}=0,\quad n=1,2,\ldots; \tag{16} \]
\[ w_{i0}=\varphi_i=\varphi(x_i),\quad i_c\le i\le N; \tag{17} \]
\[ \gamma_{n-1}\frac{h}{\tau_n}=k(y_n)\delta_x w_{i_c+n,n}-\bar k(y_{n-1})\delta_x\bar w_{i_c+n-1,n}+\Phi_{n-1},\quad n=1,2,\ldots, \tag{18} \]
where the following notation has been adopted:
\[ \bar k_i=\bar k(x_i),\quad k_i=k(x_i),\quad \bar\rho_{i,n-1}=\bar\rho(x_i,t_{n-1},\bar w_{i,n-1}), \]
\[ \rho_{i,n-1}=\rho(x_i,t_{n-1},w_{i,n-1}),\quad \bar q_{n-1}=\bar q(t_{n-1},w_{0,n-1}), \]
\[ q_{n-1}=q(t_{n-1},w_{N,n-1}),\quad \gamma_{n-1}=\gamma(y_{n-1},t_{n-1}),\quad \Phi_{n-1}=\Phi(y_{n-1},t_{n-1}), \]
\[ \delta_x z_{in}=\frac{1}{h}(z_{i+1,n}-z_{in}),\quad \delta_{\bar x} z_{in}=\frac{1}{h}(z_{in}-z_{i-1,n}),\quad \delta_t^- z_{in}=\frac{1}{\tau_n}(z_{in}-z_{i,n-1}). \]

System (10)—(18) is nonlinear with respect to the unknowns \(\bar w_{in}, w_{in}, \tau_n\), and the following iteration method is proposed for its solution. Let \(\bar w_{ik}, w_{ik}, \tau_k\) \((k=0,1,\ldots,n-1)\) \((\tau_0\ \text{is not determined})\), satisfying (10)—(18), be known. Then the successive approximations \(\bar w_{in}^{(s)}, w_{in}^{(s)}, \tau_n^{(s)}\),

For \(s=0,1,2,\ldots\), we shall determine the quantities \(\bar w_{in}, w_{in}, \tau_n\) \((n \ge 1)\) in the following way. Set \(\tau_n^{(0)}>0\), and from (10)—(13) and (14)—(17), with \(\tau=\tau_n^{(0)}\), find \(\bar w_{in}^{(0)}, w_{in}^{(0)}\), while from (19)

\[ \tau_n^{(s+1)}= \frac{1}{\Phi_{n-1}+\bar q_{n-1}-q_{n-1}} \left[ h\gamma_{n-1}+\tau_n^{(s)} \left( \bar q_{n-1}+\bar k(y_{n-1})\delta_x \bar w_{i_c+n-1,n}^{(s)} - q_{n-1}-k(y_n)\delta_x w_{i_c+n,n}^{(s)} \right) \right] \tag{19} \]

for \(s=0\) we obtain \(\tau_n^{(1)}\). In general, knowing \(\tau_n^{(s)}>0\), from (10)—(13) and (14)—(17), with \(\tau=\tau_n^{(s)}\), we find \(\bar w_{in}^{(s)}, w_{in}^{(s)}\), and then from (19) obtain \(\tau_n^{(s+1)}\), etc. In solving the systems (10)—(13) and (14)—(17) it is convenient to use the sweeping method (7).

Theorem 1. Let the following conditions be satisfied: 1) \(\bar\varphi(x)\in C\) on \([0,c]\), \(\bar\varphi(c)=0\), \(-\bar q(0,\bar\varphi(0))\le \bar k_i\delta_x\bar\varphi_i\le 0\), \(\delta_x^{-}(\bar k_i\delta_x\bar\varphi_i)\ge 0\); \(\varphi(x)\in C\) on \([c,l]\), \(\varphi(c)=0\), \(-\bar q(0,\bar\varphi(0))-\Phi(c,0)\le k_i\delta_x\varphi_i\le -q(0,\varphi(l))\), \(\delta_x^{-}(k_i\delta_x\varphi_i)\ge 0\); 2) \(\bar q(t,\bar u)\ge 0\), \(\bar q_t\ge 0\), \(\bar q_{\bar u}\ge 0\) for \(t\ge 0\), \(0\le \bar u\le l_{m_0}\bar q_{\max}/\bar k_{\min}\); \(q(t,u)\ge 0\), \(q_t\le 0\), \(q_u\le 0\) for \(t\ge 0\), \(\varphi(l)\le u\le 0\); 3) \(\bar k(x)\ge \bar k_{\min}>0\), \(\bar k(x)\in Q\) \((x\ne l_m)\), \(\bar\rho(x,t,\bar u)\ge \bar\rho_{\min}>0\), \(\bar\rho(x,t,\bar u)\in Q\) \((x\ne l_m)\) \((m=1,2,\ldots,m_0-1)\) in \(G_2=\{0\le x\le l_{m_0},\ t\ge 0,\ 0\le \bar u\le l_{m_0}\bar q_{\max}/\bar k_{\min}\}\); \(k(x)\ge k_{\min}>0\), \(k(x)\in Q\) \((x\ne l_m)\), \(\rho(x,t,u)\ge \rho_{\min}>0\), \(\rho(x,t,u)\in Q\) \((x\ne l_m)\) \((m=m_0,\ldots,m_1-1)\) in \(G_1=\{c\le x\le l,\ t\ge 0,\ \varphi(l)\le u\le 0\}\).

Then, for any choice of \(\tau_n^{(0)}>0\), the iterations (10)—(17), (19) are uniquely determined for all \(s=0,1,2,\ldots\), and, as \(s\to\infty\), the quantities \(\bar w_{in}^{(s)}, w_{in}^{(s)}, \tau_n^{(s)}\), changing monotonically, will converge to the solution of the system (10)—(18).

The proof of this theorem is carried out approximately according to the same scheme as the proof of analogous theorems in \((^{2,4,5})\).

Along with the conditions of Theorem 1, consider the following conditions:

a) \(\bar q(t,\bar u)\equiv \text{const}=\bar q\), \(q(t,u)\equiv \text{const}=q\); \(\bar k_0\delta_x\bar\varphi_0=-\bar q\), \(k_{N-1}\delta_x\varphi_{N-1}=-q\), \(\delta_x^{-}(\bar k_i\delta_x\bar\varphi_i)\le \mu\bar\rho(x_i,0,\bar\varphi_i)\), \(\delta_x^{-}(\bar k_{i_m}\delta_x\bar\varphi_{i_m})=0\) \((m=1,2,\ldots,m_0-1)\), \(\delta_x^{-}(k_i\delta_x\varphi_i)\le \mu\rho(x_i,0,\varphi_i)\), \(\delta_x^{-}(k_{i_m}\delta_x\varphi_{i_m})=0\) \((m=m_0,\ldots,m_1-1)\) for all sufficiently small \(h>0\); \(\bar\mu,\mu\) are constants;

b) \(\bar k(x)\in Q^{(5)}\) \((x\ne l_m)\), \(\bar\varphi(x)\in Q^{(2)}\) \((x\ne l_m)\), \(\bar\rho(x,t,u)\in Q^{(4,0,4)}\cap Q^{(0,1,0)}\) \((x\ne l_m)\), \(\bar\rho_t\ge 0\), \(\bar\rho_{\bar u}\ge 0\) in \(G_2\); \(k(x)\in Q^{(5)}\) \((x\ne l_m)\), \(\varphi(x)\in Q^{(2)}\) \((x\ne l_m)\), \(\rho(x,t,u)\in Q^{(4,0,4)}\cap Q^{(0,1,0)}\) \((x\ne l_m)\), \(\rho_t\ge 0\), \(\rho_u\ge 0\) in \(G_1\) \((m=1,2,\ldots,m_1-1)\);

c) \(\min(\Phi+\bar q-q)>l_{m_0-1}M_2^0\bar\rho_{\max}+(l_{m_0}-l_{m_0-1})\max\{\bar M_2^0\bar\rho_{\max};\,M_2^0\rho_{\max}\}+(l-l_{m_0})M_2\rho_{\max}\), where \(\bar M_2=\max\{\bar\mu;\Lambda_1\bar q/\bar k_{\min}\}\), \(M_2=\max\{\mu;\Lambda_1(\bar q+\Phi_{\max})/k_{\min}\}\), \(\Lambda_1=\max[(\Phi+\bar q-q)/\gamma]\);

d) \(\gamma(x,t)\in C^{(0,1)}\), \(\Phi(x,t)\in C\) in \(\Delta=\{c\le x\le l_{m_0},\ t\ge 0\}\).

If the conditions of Theorem 1 and conditions a)—d) are fulfilled, there is a sequence \(h_\nu\), \(\nu=1,2,\ldots\), such that the polygonal lines \(y(t,h_\nu)\), obtained by joining by straight-line segments the points \((y_n,t_n)\), as \(\nu\to\infty\), converge uniformly on some interval \(0\le t\le T\) to a monotone curve \(y(t)\), \(c\le y(t)\le l_{m_0}\), and there also exist solutions \(\bar u(x,t)\), \(u(x,t)\) of the boundary-value problems (1)—(4) and (5)—(8) for this curve \(y(t)\). Suppose the inequalities

e) \(|u_t(x',t)-u_t(x'',t)|\le M(\delta,\gamma)|x'-x''|^\alpha\) in \(\{y(t)+\delta\le x',x''\le l-\delta,\ \gamma\le t\le T\}\) and \(|u_t(x',t)-u_t(x'',t)|\le M(\delta,\gamma)|x'-x''|^\alpha\) in \(\{\delta\le x',x''\le y(t)-\delta,\ \gamma\le t\le T\}\), where \(l_m\le x',x''\le l_{m+1}\) \((m=0,1,\ldots,m_1-1)\), for all sufficiently small \(\delta>0,\ \gamma>0\); \(0<\alpha=\alpha(\delta,\gamma)\le 1\).

Theorem 2. If the hypotheses of Theorem 1 and conditions a)—e) are satisfied, then there exists a solution of problem (1)—(9), which can be obtained as the limit, as \(h_\nu \to 0\), of solutions of the difference problem (10)—(18).

Using Theorem 2 and the method and results of papers \((^{8-10})\), one can prove the following two theorems:

Theorem 3. If \(\bar{\rho}=\bar{\rho}(x,t)\), \(\rho=\rho(x,t)\) and all the hypotheses of Theorem 1 and conditions a)—d) are satisfied, then there exists a solution of problem (1)—(9).

Theorem 4. If all the hypotheses of Theorem 1 and conditions a)—d) are satisfied with the classes \(Q\) replaced by the classes \(C\) (i.e. \(m_1=m_0=1\)), then there exists a solution of the Stefan-type problem (1)—(9).*

\(2^\circ\). Consider the one-phase Stefan-type problem: to find the functions \(u(x,t)\), \(y(t)\) from conditions (5)—(8) and condition \((1^3,{}^5)\) (cf. (11)):

\[ \gamma(y(t),t)y'(t)=k(y(t))u_x(y(t),t)+\Phi(y(t),t),\qquad y(0)=c. \tag{20} \]

For an approximate solution of problem (5)—(8), (20), one may use the difference scheme (14)—(18) and the iteration method, putting in (18) and (19)
\(\bar{w}_{in}\equiv\bar{w}_{in}^{(s)}\equiv\bar{q}\equiv0\). Theorems analogous to Theorems 1—4 hold; their statements may be obtained from the corresponding Theorems 1—4 by putting in them
\(\bar{q}\equiv\bar{\varphi}\equiv\bar{u}\equiv\bar{w}_{in}\equiv\bar{w}_{in}^{(s)}\equiv\bar{M}_2\equiv0\). We note that, in problem (5)—(8), (20), the condition \(u(y(t),t)\equiv0\) may be replaced by the more general one:
\(u(y(t),t)=\psi(y(t),t)\), where \(\psi(x,t)\) is a prescribed function.

\(3^\circ\). In an analogous way, another one-phase Stefan-type problem is considered: to find the functions \(\bar{u}(x,t)\), \(y(t)\) from conditions (1)—(4) and \((^5,{}^6)\) (cf. \((^{12,13})\)):

\[ \gamma(y(t),t)y'(t)=-\bar{k}(y(t))\bar{u}_x(y(t),t)+\Phi(y(t),t),\qquad y(0)=c. \tag{21} \]

\(4^\circ\). In contrast to papers \((^{1-5,12,13})\), the uniqueness of the solutions of all the Stefan-type problems considered above in the class of sufficiently smooth solutions, for sufficiently smooth coefficients of the equations, can be proved without the assumption of monotonicity of \(y(t)\). In particular, the following is true.

Theorem 5. If \(\bar{q}=\bar{q}(t)\), \(q=q(t)\), \(\bar{\rho}=\bar{\rho}(x,t)\), \(\rho=\rho(x,t)\), \(\Phi(x,t)\in C^{(1,0)}\) in \(\Delta\), and conditions b) and d) are satisfied, then problem (1)—(9) has no more than one solution.

The author expresses deep gratitude to B. M. Budak for proposing the problem and for attentive guidance in the execution of the present work, and also to A. B. Uspenskii for useful discussion of the results.

Moscow State University
named after M. V. Lomonosov

Received
8 II 1964

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* We note that analogous theorems from \((^{4,5})\), which are a special case of Theorem 4, were proved under the additional assumption of boundedness of the fourth derivatives of the solutions \(\bar{u}(x,t)\), \(u(x,t)\) with respect to the variable \(x\).