UDC 517.946

MATHEMATICS

V. N. GOL'DBERG

THE OCCURRENCE OF DISCONTINUITIES IN THE CONTINUATION OF SOLUTIONS OF NONLINEAR MIXED PROBLEMS FOR THE STRING EQUATION

(Presented by Academician I. G. Petrovskii on 26 XII 1966)

1°. In the present note the following nonlinear mixed problem is considered:

\[ u_{xx}-u_{tt}=-\Phi(x,t,u); \tag{1} \]

\[ u(x,0)=\varphi(x),\qquad u_t(x,0)=\psi(x)\quad \text{for } 0\le x\le 1; \tag{2} \]

\[ a_0(u)u_x+b_0(u)u_t=f_0(t,u)\quad \text{for } x=0; \tag{3} \]

\[ a_1(u)u_x+b_1(u)u_t=f_1(t,u)\quad \text{for } x=1. \tag{4} \]

In \((^1,^2)\) it was shown that if the function \(u\in C_2^0(\overline{\Pi}_{T_0})\) is a solution of problem (1)—(4) in the rectangle
\[ \Pi_{T_0}=\{0\le x\le 1,\ 0\le t\le T_0\}\quad (0<T_0<+\infty) \]
and

\[ h_i(u(i,T_0))\equiv b_i(u(i,T_0))-(-1)^i a_i(u(i,T_0))\ne 0 \quad (i=0,1), \tag{5} \]

then the solution \(u\) can be continued uniquely in the rectangle \(\Pi_{T_0+\Delta T}\) \((\Delta T>0)\) with preservation of smoothness. Inequality (5) is an unnatural restriction in a number of physical problems (see, for example, \((^3)\)). Below the solvability of problem (1)—(4) is investigated in the case when, upon continuation in \(t\) of a smooth solution, inequality (5) for \(i=0\) becomes an equality at some moment \(t=T^*\ne+\infty\). It turns out that for \(t>T^*\) the solution of problem (1)—(4) becomes nonunique, and discontinuities and loss of smoothness of solutions arise; moreover, increasing the smoothness and compatibility of the data of the problem (the functions \(\Phi,\varphi,\psi,a_i,b_i,f_i\)) does not lead to unique continuability of the solution and to preservation of smoothness for \(t>T^*\). In this connection, in the present work discontinuous and piecewise-smooth solutions of problem (1)—(4) are constructed for \(t>T^*\). For the examples given in §4 it is shown that the discontinuous and piecewise-smooth solutions introduced below arise naturally if one studies the behavior of solutions of mixed problems obtained under small perturbations of the operator \(a_0(u)u_x+b_0(u)u_t\) and of the initial conditions, as the perturbations tend to zero.

Choose an arbitrary \(0<T<1\). Put
\[ R^1=(-\infty,+\infty),\qquad D_T^0=\overline{\Pi}_T\times R^1,\qquad D_T^1=[0,T]\times R^1. \]

Let:

1) the functions \(\Phi\in C_1(D_1^0)\), \(f_i\in C_1(D_1^1)\), \(a_i,b_i\in C_1(R^1)\), \(\varphi\in C_2[0,1]\), \(\psi\in C_1[0,1]\);

2) the function \(h_0\) have only isolated and simple zeros on the interval \(R^1\), while the function \(h_1\) have no zeros on the interval \(R^1\).

Suppose that in the rectangle
\[ \Pi_{T^*}=\{0\le x\le 1,\ 0\le t<T^*\}, \]
where \(0<T^*<1\),* there exists a solution \(u\in C_2^0(\Pi_{T^*})\) of problem (1)—(4), and moreover
\[ \sup_{\Pi_{T^*}} |u|<+\infty,\qquad h_0(u(0,t))\ne 0\quad \text{for } 0\le t<T^*,\qquad h_0(u(0,t))\to 0 \]
as \(t\to T^*\).

* Obviously, the inequality \(0<T^*<1\) does not restrict the generality.

Lemma 1. There exists \(\lim_{t\to T^*} \overset{0}{u}(0,t)=u^*\), and \(h_0(u^*)=0\).

2°. In the domain \(D_{T^*}^1\) define the function

\[ P(t,u)=f_0(t,u)-a_0(u)[\varphi'(t)+\psi(t)]-a_0(u)\int_0^t \Phi(t-\tau,\tau,u(t-\tau,\tau))\,d\tau . \]

It can be shown that \(P\in C_1(D_{T^*}^1)\). Put
\[ P_0=P(T^*,u^*),\quad P_{1,0}=P_t'(T^*,u^*),\quad P_{0,1}=P_u'(T^*,u^*), \]
\[ \overset{0}{Q}_\delta=\{0\le x\le 1,\ 0\le t\le \min(x+T^*-\delta,T^*)\},\quad 0<\delta\le T^*. \]
Note that \(|P_0|<+\infty\).

Theorem 1. Whatever \(P_0\) may be, \(\overset{0}{u}\in C(\overline{\Pi}_{T^*})\).*

Theorem 2. Let \(P_0\ne 0\). Then \(\overset{0}{u}\in C_2(\overline{Q}_\delta)\) for \(\delta>0\),
\[ \sup_{\overline{Q}_\delta}|D^1u|\to\infty \]
as \(\delta\to 0\), and
\[ \sup_{\Pi_{T^*}}\left|D^1u(x,t)\right|\sqrt{\left|t-(x+T^*)\right|}<+\infty . \]

One can construct examples of nonlinear mixed problems with arbitrarily smooth and well-compatible data for which Theorem 2 holds.

Theorem 3. Let \(P_0=0,\ P_{1,0}\ne 0\). Then \(\overset{0}{u}\in C_2(\overline{Q}_\delta)\) for \(\delta>0\) and \(\overset{0}{u}\in C_1(\overline{\Pi}_{T^*})\).

Choose an arbitrary \(T^*<T<1\). Put
\[ \Omega_T=\{(x,t)\in \Pi_T,\ t\ne x+T^*\}, \]
\[ G_T^0=\{0\le x<T-T^*,\ x+T^*<t\le T\}, \]
\[ G_T^1=\Omega_T\setminus G_T^0. \]

Let \(\mathfrak{M}_T\) be the set of functions \(u\in C(\overline{\Pi}_T)\) such that:

1) \(u\in C_1(\overline{G}_T^{\,j})\) \((j=0,1)\), \(u_x+u_t\in C(\overline{\Pi}_T)\), and, whatever closed domain \(\overline{D}\subseteq \Omega_T\) may be, the function \(u\in C_2(\overline{D})\);

2) \(u\equiv \overset{0}{u}\) in the rectangle \(\overline{\Pi}_{T^*}\).

A function \(u\in\mathfrak{M}_T\) will be called a solution of problem (1)—(4) in the rectangle \(\overline{\Pi}_T\) if it satisfies equation (1) in the domain \(\Omega_T\) and equations (3), (4) for \(0\le t\le T\).**

Theorem 4. Let \(P_0=0,\ P_{1,0}\ne 0\), and \(P_{1,0}h_0'(u^*)>0\). Then there exists a \(T^*<T<1\) such that in the set \(\mathfrak{M}_T\) there exist two and only two solutions \(u_i\) \((i=1,2)\) of problem (1)—(4) in the rectangle \(\overline{\Pi}_T\), moreover
\[ u_1\in C_1(\overline{\Pi}_T),\qquad u_2\in C_1(\Pi_T). \]

Theorem 5. Let \(P_0=0,\ P_{1,0}\ne 0,\ P_{1,0}h_0'(u^*)<0\)*** and
\[ P_{0,1}^2+4P_{1,0}h_0'(u^*)\ne 0. \]
Then there exists a \(T^*<T<1\) such that in the set \(\mathfrak{M}_T\) there exists an infinite set of solutions of problem (1)—(4).

One can construct examples of nonlinear mixed problems with arbitrarily smooth and well-compatible data for which Theorems 4 and 5 hold.

3°. Below, discontinuous solutions of problem (1)—(4) are constructed for \(t>T^*\) for any \(P_0\). The introduction of discontinuous solutions in the case \(P_0\ne 0\) is caused by the absence of continuous solutions in the rectangle \(\overline{\Pi}_T\) for any \(T>T^*\). Indeed, independently of the value of \(P_0\), for any \(T\) \((0<T\le 1)\) one can define the notion of a continuous generalized solution (c.g.s.) of problem (1)—(4) in the rectangle \(\overline{\Pi}_T\). The function \(\overset{0}{u}\) is the unique c.g.s. in the rectangle \(\overline{\Pi}_T\) for any \(0<T\le T^*\), and in the case \(P_0=0\), for the corresponding \(T\), all solutions are such,

* That is, the function \(u\) admits a continuous prolongation as \(t\to T^*\).

** The derivatives \(D^1u\) at the point \((0,T^*)\) should be understood as the corresponding one-sided derivatives.

*** It can be shown that if \(P_0=0,\ P_{1,0}\ne 0\), and \(P_{1,0}h_0'(u^*)<0\), then \(P_{0,1}\ne 0\).

specified in Theorems 4 and 5. It can be shown that in the case \(P_0 \ne 0\) problem (1)—(4) does not have a n.g.s. in the rectangle \(\overline{\Pi}_T\) for any \(T > T^*\), and increasing the smoothness and compatibility of the data of the problem does not lead to the existence of a n.g.s. The advisability of introducing discontinuous solutions in the case \(P_0 = 0\) is indicated by Example 2 given in §4.

Put

\[ H_0(u)=\int_{\varphi(0)}^u h_0(\xi)\,d\xi,\qquad J(t;u)=\int_0^t P(\tau,u(0,\tau))\,d\tau . \]

Lemma 2. Whatever \(P_0\) may be,

\[ H_0(u(0,t))=J(t;u) \qquad \text{for } 0\le t\le T^* . \]

Suppose there exists at least one value \(\overline{u}^{\,*}\) different from \(u^*\) such that

\[ H_0(\overline{u}^{\,*})=J(T^*;u),\qquad H_0'(\overline{u}^{\,*})=h_0(\overline{u}^{\,*})>0 . \]

Fix an arbitrary such value \(\overline{u}^{\,*}\). Choose any \(T^*<T<1\).

Denote by \(\mathfrak{R}_T(\overline{u}^{\,*})\) the set of functions such that: 1) \(u\in C_2(\overline{G_T^0})\), \(u\in C(\overline{G_T^1})\), \(u(0,T^*+0)=\overline{u}^{\,*}\); 2) \(u\equiv \overset{0}{u}\) in the rectangle \(\overline{\Pi}_{T^*}\).

We note that a function \(u\in \mathfrak{R}_T(\overline{u}^{\,*})\) has a discontinuity of the first kind on the line \(t=x+T^*\) in a neighborhood of the point \((0,T^*)\).

Theorem 6. Let \(P_0=0\). Then there exists \(T^*<T<1\) such that in the set \(\mathfrak{R}_T(\overline{u}^{\,*})\) there exists a unique function \(u\in C_1(\overline{G_T^1})\) which satisfies equation (1) in the domain \(\Omega_T\) and equations (3), (4) for \(0\le t\le T\).

Theorem 7. Let \(P_0\ne 0\). Then there exists \(T^*<T<1\) such that in the set \(\mathfrak{R}_T(\overline{u}^{\,*})\) there exists a unique function \(u\) having the following properties:

1) \(|D^1u(x,t)|\to\infty\) as \(t-(x+T^*)\to -0\),

\[ \sup_{G_T^1}|D^1u(x,t)|\sqrt{|t-(x+T^*)|}<+\infty . \]

2) the function \(u\) satisfies equation (1) in the domain \(\Omega_T\) and equations (3), (4) for \(t\in[0,T^*)\), \(t\in[T^*,T]\), and \(t\in[0,T]\), respectively.

§4. In the examples given below the following nonlinear mixed problem is considered

\[ u_{xx}=u_{tt}; \tag{6} \]

\[ u(x,0)=0,\qquad u_t(x,0)=\psi(x)\qquad \text{for } 0\le x\le 1; \tag{7} \]

\[ -u_x+\frac{\partial f(u)}{\partial u}\,u_t=0 \qquad \text{for } x=0; \tag{8} \]

\[ u_x=0 \qquad \text{for } x=1, \tag{9} \]

where \(f(u)=\frac{5}{3}u^3-\frac{5}{2}u^2+\frac{11}{6}\), and \(\psi(x)\) is a function of class \(C_1[0,1]\), chosen in each example in a special way and satisfying the compatibility conditions \(\psi^{(s)}(j)=0\) \((s,j=0,1)\). Note that \(h_1(u)\equiv 1\), while the equation \(h_0(u)=0\) has two real roots

\[ u_j^*=\frac12+(-1)^j\frac{\sqrt5}{10}\qquad (j=1,2). \]

Example 1. Choose an arbitrary function \(\psi\) such that: a) \(\psi(x)>0\) for \(0<x\le \frac34\); b) the function

\[ \sigma(x)=f(0)+\int_0^x \psi(\xi)\,d\xi \]

is such that

\[ \sigma(x)<u_1^*+f(u_1^*) \quad \text{for } 0<x<\frac12 \]

and

\[ \sigma\!\left(\frac12\right)=u_1^*+f(u_1^*) . \]

Theorem 8. Problem (6)—(9) has in the rectangle \(\Pi_{1/2}\) a unique solution \(u\in C_2(\Pi_{1/2})\), moreover

\[ h_0(u(0,t))>0 \quad \text{for } 0\le t<\frac12, \]

\[ u(0,t)\to u_1^* \quad \text{as } t\to \frac12,\qquad P_0\ne 0 . \]

In the example under consideration \(T^*=\frac12\). It can be shown that the equation

\[ H_0(u)=J(T^*;u) \]

has a unique root \(\overline{u}^{\,*}\), different from the root \(u_1^*\). Consequently, by Theorem 7, there exists \(T^*<T<1\) such that the problem

(6)—(9) has in the rectangle \(\Pi_T\) a unique discontinuous solution
\(u_p \in \mathfrak{R}_T(\bar u^*)\). Let us now consider equation (6) with the initial conditions (7), the boundary condition (9), and the following boundary condition:

\[ \mu u_{tt}-u_x+\frac{\partial f(u)}{\partial u}\,u_t=0 \quad (\mu>0)\quad \text{for } x=0. \tag{10} \]

Theorem 9. There exists a \(\mu_0>0\) such that, for any \(0<\mu<\mu_0\), problem (6), (7), (9), (10) has in the rectangle \(\overline{\Pi}_T\) a unique solution \(u_\mu\in C_2(\overline{\Pi}_T)\), and \(u_\mu\to u_p\) as \(\mu\to0\) uniformly with respect to \(x,t\) in each closed domain \(\bar D\subset\Omega_T\).

Example 2. Choose an arbitrary function \(\psi\) satisfying condition b) and such that \(\psi(x)>0\) for \(0\le x<1/2\), \(\psi(1/2)=0\), \(\psi'(1/2)<0\).

Theorem 10. In the rectangle \(\Pi_{1/2}\) there exists a unique solution \(u\in C_2(\Pi_{1/2})\) of problem (6)—(9), and \(h_0(u(0,t))>0\) for \(0\le t<1/2\), \(u(0,t)\to u_1^*\) as \(t\to1/2\); \(P_0=0\), \(P_{1,0}\ne0\), \(P_{1,0}h_0'(u_1^*)>0\).

As in Example 1, \(T^*=1/2\).

Theorem 11. Let \(T\) be the number indicated in Theorem 4. For any \(\varepsilon>0\), one can specify a \(\mu_0(\varepsilon)>0\) such that, for any \(0<\mu<\mu_0(\varepsilon)\), problem (6), (7), (9), (10) has in the rectangle \(\overline{\Pi}_T\) a unique solution \(u_\mu\in C_2(\overline{\Pi}_T)\), and

\[ \max_{\overline{\Pi}_T}|u_\mu-u_2|<\varepsilon, \]

where \(u_2\) is the solution of problem (6)—(9) appearing in Theorem 4.

As in Example 1, the existence is established of such a \(T^*<\bar T<1\) that in the rectangle \(\overline{\Pi}_{\bar T}\) problem (6)—(9) has a unique discontinuous solution \(u_p\in\mathfrak{R}_{\bar T}(\bar u^*)\), where \(\bar u^*\) is the unique root, distinct from \(u_1^*\), of the equation \(H_0(u)=J(T^*;u)\).

Theorem 12. Whatever closed domain \(\bar D\subset G^1_{\bar T}\) is given, for any \(\varepsilon>0\) one can specify a \(\delta_0(\varepsilon)>0\) such that, for any \(0<\delta<\delta_0(\varepsilon)\), there exist \(0<T^*(\delta)<T^*\) and a function \(\psi_\delta(x)\in C_1[0,1]\) satisfying the relations

\[ \bar D\subset\Gamma_\delta=\{0\le x\le1,\ 0\le t\le \min(x+T^*(\delta),\bar T)\}, \]

\[ \sum_{s=0,1}\max_{0\le x\le1}|\psi_\delta^{(s)}(x)-\psi^{(s)}(x)|<\delta \]

and such that problem (6)—(9), with initial conditions \(u(x,0)=0\), \(u_t(x,0)=\psi_\delta(x)\):

1) has in the rectangle \(\overline{\Pi}_{T^*(\delta)}\) a unique solution \(u^\delta\in C_2(\overline{\Pi}_{T^*(\delta)})\), with \(h_0(u^\delta(0,t))>0\) for \(0\le t<T^*(\delta)\), \(u^\delta(0,t)\to u_1^*\) as \(t\to T^*(\delta)\); \(P_0\ne0\);

2) has in the rectangle \(\overline{\Pi}_{\bar T}\) a unique discontinuous solution
\(u_p^\delta\in\mathfrak{R}_{\bar T}(\bar u_\delta^*)\)*, where \(\bar u_\delta^*\) is the unique root, distinct from \(u_1^*\), of the equation \(H_0(u)=J(T^*(\delta);u^\delta)\), and

\[ \max_{\Gamma_\delta}|u_R-u_p^\delta|<\varepsilon,\qquad \max_{\substack{0\le x\le \bar T-T^*\\ x+T^*\le t\le \bar T}}|u_p-u_p^\delta|<\varepsilon. \]

Scientific Research Radiophysics Institute
at Gorky State University
named after N. I. Lobachevsky

Received
22 XII 1966

CITED LITERATURE

1. V. E. Abolina, A. D. Myshkis, Matem. sborn., 50 (92), 4, 423 (1960).
2. V. N. Golubev, Yu. I. Neimark, Matem. sborn., 67 (109), 1, 16 (1965).
3. A. A. Witt, ZhTF, 6, 9, 1459 (1936).

* Here, in the definition of the domains \(\Omega_T\), \(G^i_T\) \((i=0,1)\) and the set \(\mathfrak{R}_{\bar T}(\bar u_\delta^*)\), one should put \(T^*=T^*(\delta)\), \(\bar u^*=\bar u_\delta^*\), \(u=u^\delta\).