UDC 517.946

Mathematics

V. N. Gol’dberg

The Occurrence of Discontinuities in the Continuation of Solutions of Nonlinear Mixed Problems for Hyperbolic Equations in the Plane

(Presented by Academician I. G. Petrovsky, March 6, 1968)

1°. In the strip \(0 \le x \le 1,\ 0 \le t < +\infty\) consider the mixed problem

\[ u_{xx}-u_{tt}=A(x,t,u)u_x+B(x,t,u)u_t+F(x,t,u), \tag{1} \]

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

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

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

It is known \((^{1,2})\) that if the function \(\dot u\in C_2(\overline{\Pi}_{T_0})\) is a solution of problem (1)—(4) in the rectangle
\[ \overline{\Pi}_{T_0}=\{0\le x\le 1,\ 0\le t\le T_0\},\qquad 0<T_0<+\infty, \]
and

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

then the solution \(u\) is continued uniquely into \(\overline{\Pi}_{T_0+\Delta T}\) \((\Delta T>0)\) with preservation of smoothness. Below, problem (1)—(4) is studied in the case when, under continuation of a solution of class \(C_2\), inequality (5) for \(i=0\) turns into an equality at some time \(t=T^*,\ 0<T^*<1/2\).* The violation of inequality (5) is connected with the occurrence of discontinuous oscillations in a telegraph line, first studied by A. A. Vitt \((^4)\).

In the present paper a definition is introduced and theorems of existence and uniqueness of a discontinuous solution (d.s.) of problem (1)—(4) for \(t>T^*\) are formulated. Theorem 5 (see 4°) shows that the d.s. of problem (1)—(4) arises in a natural way if one studies the behavior, as \(\mu\to 0\), of the solutions of mixed problems obtained by perturbing the operator \(a_0(\dot u)u_x+b_0(u)u_t\) by the term \(\mu u_{tt}\) \((\mu>0)\). The necessity of introducing a d.s. for \(t>T^*\) is caused by the following circumstance. Consider problem (1)—(4) in \(\Pi_{1/2}\). Passing, by the substitution \(v=u_x+u_t,\ w=u_x-u_t\), to the mixed problem for the corresponding hyperbolic system and integrating it along characteristics, one can obtain an integral equation for the function \(u(x,t)\), equivalent to problem (1)—(4). Such an equation leads to the natural definition of the concept of a continuous generalized solution (c.g.s.) of problem (1)—(4) in \(\overline{\Pi}_T,\ 0<T\le 1/2\). In this case \(u\) turns out to be the unique c.g.s. of problem (1)—(4) in \(\overline{\Pi}_{T^*}\). However, in the “general” case studied here,** for any \(\Delta T>0\) there is no c.g.s. of problem (1)—(4) in \(\overline{\Pi}_{T^*+\Delta T}\), and any increase in the smoothness and compatibility of the data of the problem (the functions \(A,B,F,a_i,b_i,f_i,\varphi,\psi\)) does not lead to the existence of a continuous solution for \(t>T^*\).

Under the assumption \(A=B=0\), a d.s. was considered by us in \((^3)\).

2°. Statement of the problem. Smoothness of the function \(\dot u\) in \(\overline{\Pi}_{T^*}\).
1. Put \(R_1=(-\infty,+\infty)\), \(D_0=\Pi_{1/2}\times R_1\), \(D_1=[0,1/2]\times R_1\), \(h_i=b_i+(-1)^{i+1}a_i\). Let:

\[ \underline{\phantom{h_i=b_i+(-1)^{i+1}a_i.\ \text{Let:}}} \]

* The inequality \(0<T^*<1/2\) involves no loss of generality.
** We have in mind the restrictions on the function \(h_0\) given in 2°, and inequality (10).

1) \(A,\ B \in C_2(D_0),\ F \in C_1(D_0),\ f_i \in C_1(D_1),\ a_i,\ b_i \in C_1(R_1),\ \varphi \in C_2[0,1],\ \psi \in C_1[0,1]\);

2) \(h_0\) has only isolated and simple zeros on \(R_1\);

3) \(h_1\) has no zeros on \(R_1\).

Let, in the rectangle \(\Pi_{T^*}=\{0 \le x \le 1,\ 0 \le t < T^*\}\), there exist a solution \(\mathring u \in C_2(\Pi_{T^*})\) of problem (1)—(4), and

\[ \text{1) }\ \sup_{\Pi_{T^*}}|\mathring u|<+\infty; \tag{6} \]

2) \(h_0(\mathring u(0,t))\ne 0\) for \(0 \le t<T^*\); without loss of generality we shall assume that

\[ h_0(\mathring u(0,t))>0 \quad \text{for } 0 \le t<T^*; \tag{7} \]

\[ \text{3) }\ \inf_{0\le t<T^*} h_0(\mathring u(0,t))=0. \tag{8} \]

Inequality (6) means that the “graph of the function \(\mathring u\)” for \((x,t)\in \Pi_{T^*}\) belongs to the domain of definition of the coefficients and free terms of equations (1), (3), (4); however, equality (8) excludes the possibility of applying, for \(t\le T^*\), the uniqueness and continuation theorems for a solution with preservation of smoothness\(^*\), established in (2).

Lemma 1. There exists

\[ \lim_{t\to T^*}\mathring u(0,t)=u^* \tag{9} \]

and \(h_0(u^*)=0\).

By virtue of Lemma 1 and inequality (7), either \(\mathring u(0,t)<u^*\) or \(\mathring u(0,t)>u^*\) for \(0\le t<T^*\). Without loss of generality we shall assume that \(\mathring u(0,t)<u^*\).

1. For \((x,t)\in\Pi_{T^*}\) put \(\mathring v=\mathring u_x+\mathring u_t\).

Lemma 2. The functions \(\mathring u,\mathring v\in C(\overline{\Pi}_{T^*})\)\(^{**}\).

Let

\[ \Gamma^0=f_0(T^*,u^*)-a_0(u^*)\mathring v(0,T^*)\ne 0. \tag{10} \]

Denote by \(\overline{\Pi}'_{T^*}\) the rectangle \(\overline{\Pi}_{T^*}\) with the point \((0,T^*)\) “punctured.”

Theorem 1. The function \(\mathring u\in C_2(\overline{\Pi}'_{T^*})\) and

\[ |\mathring u_x(x,t)|\to +\infty,\qquad |\mathring u_t(x,t)|\to +\infty, \quad \text{if } \ t-(x+T^*)\to 0, \]

\[ \sup_{\overline{\Pi}'_{T^*}}|\mathring u_x(x,t)|\sqrt{|t-(x+T^*)|}<+\infty, \]

\[ \sup_{\overline{\Pi}'_{T^*}}|\mathring u_t(x,t)|\sqrt{|t-(x+T^*)|}<+\infty. \]

The properties of the derivatives \(\mathring u_x,\mathring u_t\) stated in Theorem 1 are determined by the properties of the function \(h_0\), and not by the smoothness and compatibility of the data of problem (1)—(4). Theorem 1 holds even in the case of analytic and arbitrarily well compatible data.

3°. Discontinuous solution of problem (1)—(4).

1. For \(T^*<T\le \tfrac12\) put \(K_T^1=\{0\le x\le t-T^*,\ T^*\le t\le T\}\), \(K_T^0=\overline{\Pi}_T\setminus K_T^1\). Let \(\mathfrak A_T,\ T^*<T\le \tfrac12\), be the set of functions \(u(x,t)\) such that \(u=\mathring u\) in \(\Pi_{T^*}\), \(u\in C(K_T^0)\), \(u\in C_2(K_T^0)\), and

\[ \sup_{K_T^0}|u_x(x,t)|\sqrt{|t-(x+T^*)|}<+\infty, \]

\[ \sup_{K_T^0}|u_t(x,t)|\sqrt{|t-(x+T^*)|}<+\infty. \]

\(^*\) It can be shown that if \(\inf_{0\le t<T^*} h_0(\mathring u(0,t))\ne 0\), then in \(\overline{\Pi}_{T^*+\Delta T}\), for some \(\Delta T>0\), there exists a unique solution of problem (1)—(4) of class \(C_2(\overline{\Pi}_{T^*+\Delta T})\).

\(^{**}\) That is, the functions \(\mathring u,\mathring v\) admit a continuous extension as \(t\to T^*\).

Definition 1. A function \(u \in \mathfrak{R}_T,\ T^* < T \leq 1/2,\) is called a solution of problem (1)—(4) in \(K_T^0\) if \(u\) satisfies equation (1) in \(K_T^0\) and equation (4) for \(T^* \leq t \leq T\).

Lemma 3. The following alternative holds:

A. The solution \(u\) of problem (1)—(4) exists and is unique in \(K_{1/2}^0\).

B. There exists a \(T^* < T_0 \leq 1/2\) such that the solution \(u\) exists and is unique in \(K_{T_0-\varepsilon}^0\) for any \(\varepsilon > 0\), and

\[ \max_{K_{T_0-\varepsilon}^0} |\breve u| + \sup_{K_{T_0-\varepsilon}^0} |\breve u_x(x,t)|\sqrt{|t-(x+T^*)|} + \sup_{K_{T_0-\varepsilon}^0} |\breve u_t(x,t)|\sqrt{|t-(x+T^*)|} \to +\infty \quad \text{as } \varepsilon \to 0 . \]

Put \(T_1=1/2\) in case A and \(T_1=T_0-\varepsilon_0\), where \(0<\varepsilon_0<T_0\) is any fixed number, in case B.

Lemma 4. The function \(\breve v=\breve u_x+\breve u_t \in C(\breve K_{T_1}^0)\).

Below the functions \(\breve u,\breve v\) are denoted as before by \(\dot u,\dot v\).

1. Let

\[ H_0(u)=\int_{\varphi(0)}^{u} h_0(\xi)\,d\xi, \qquad I^*=\int_0^{T^*} \bigl[ f_0(\tau,\dot u(0,\tau)) - a_0(\dot u(0,\tau))\dot u(0,\tau) \bigr]\,d\tau . \]

Lemma 5. The value \(u=\bar u^*\) is a root of the equation

\[ H_0(u)=I^* . \tag{11} \]

Suppose that equation (11) has a real root \(u^*<\bar u^*<+\infty\) such that

\[ h_0(\bar u^*)>0; \qquad H_0(u)\ne I^* \quad \text{for } u^*<u<\bar u^* . \]

Denote by \(\mathfrak{R}_T,\ T^*<T\leq T_1,\) the set of functions \(u\in C_2(K_T^1)\) satisfying the condition \(u(0,T^*)=\bar u^*\).

Definition 2. A function \(u\in\mathfrak{R}_T,\ T^*<T\leq T_1,\) is called a solution of problem (1)—(4) in \(K_T^1\) if:

1) \(u\) satisfies equation (1) in \(K_T^1\) and equation (3′) for \(T^*\leq t\leq T\);

2) for \(T^*\leq t\leq T\) the relations

\[ h_0(u(0,t))>0, \]

\[ v(t-T^*,t)-P(t-T^*,t,u(t-T^*,t)) = \]

\[ = \dot v(t-T^*,t)-P(t-T^*,t,\dot u(t-T^*,t)), \]

hold, where

\[ v=u_x+u_t,\qquad P(x,t,u)=\frac12\int_0^u [A(x,t,\eta)-B(x,t,\eta)]\,d\eta . \]

Let the function \(u\in\mathfrak{R}_T,\ T^*<T\leq T_1,\) be a solution of problem (1)—(4) in \(K_T^1\).

Definition 3. The function \(u_p=u\), if \((x,t)\in K_T^1\), and \(u_p=\dot u\), if \((x,t)\in K_T^0\), is called an r.r. solution of problem (1)—(4) in \(\Pi_T\).

Theorem 2. The following alternative holds:

1) in \(K_{T_1}^1\) there exists a unique solution \(u\) of problem (1)—(4), and

\[ h_0(u(0,t))>0 \quad \text{for } T^*\leq t\leq T_1; \]

2) there exists a \(T^*<\widetilde T\leq T_1\) such that for any \(\varepsilon>0\) in \(K_{\widetilde T-}^{1}\) there exists a unique solution \(u\) of problem (1)—(4), and

\[ h_0(u(0,t))>0 \quad \text{for } T^*\leq \]

\(\leqslant t \leqslant \widetilde T-\varepsilon;\) moreover:

a) either
\[ \sup_{\substack{T^*\leqslant t<\widetilde T\\ 0\leqslant x\leqslant t-T^*}} |u|<+\infty \]
and there exists
\[ \lim_{t\to\widetilde T} u(0,t)=u^{**},\qquad h_0(u^{**})=0; \]

b) or
\[ \sup_{\substack{T^*\leqslant t<\widetilde T\\ 0\leqslant x\leqslant t-T^*}} |u|=+\infty . \]

It is not difficult to construct examples showing that all the cases indicated in Theorem 2 are in fact realized.

Put \(T_2=T_1\) in case 1) of Theorem 2, and \(T_2=\widetilde T-\varepsilon_0\), where \(0<\varepsilon_0<\widetilde T\) is any fixed number, in case 2).

Theorem 3. In \(\overline\Pi_{T_2}\) there exists a unique d.r. \(u_p\) of problem (1)—(4), and for \(T^*\leqslant t\leqslant T_2\)
\[ u(t-T^*,t)-\dot u(t-T^*,t)=\overline u^*-u^*+ \]
\[ +\int_{T^*}^{t}\left[P(\tau-T^*,\tau,u(\tau-T^*,\tau))-P(\tau-T^*,\tau,\dot u(\tau-T^*,\tau))\right]\,d\tau . \]

4°. Small perturbations of equation (3) and a discontinuous solution. Following A. A. Vitt \((^4)\), consider in \(\overline\Pi_{T_2}\) the mixed problem \((1')\)—\((4')\), defined by equations (1), (2), (4) and the boundary condition
\[ \mu u_{tt}+a_c(u)u_x+b_0(u)u_t=f_0(t,u)\qquad \text{for }x=0. \tag{3'} \]

We note that the functions \(\varphi,\psi\), generally speaking, do not satisfy the compatibility condition of problem \((1')\)—\((4')\) for \(x=0\). Consequently, for any \(T>0\) in \(\overline\Pi_T\) there is no solution of problem \((1')\)—\((4')\) of class \(C_2(\Pi_T)\).

For \(0<T\leqslant T_2\) put \(G_T^0=\{(x,t),\,0\leqslant x\leqslant T\}\), \(G_T^1=\overline\Pi_T\setminus G_T^0\). We now introduce the following

Definition 4. A function \(u_\mu(x,t)\) is called a solution of problem \((1')\)—\((4')\) in \(\overline\Pi_T\), \(0<T\leqslant T_2\), if \(u_\mu\in C_1(\overline\Pi_T)\), \(u_\mu\in C_2(\overline G_T^i)\) \((i=0,1)\), and \(u_\mu\) satisfies equation (1) in the domains \(\overline G_T^i\) and equations (2), \((3')\), (4) for \(0\leqslant x\leqslant 1,\;0\leqslant t\leqslant T\).

Theorem 4. One can specify a \(\mu^*>0\) such that, for \(0<\mu<\mu^*\), in \(\overline\Pi_{T_2}\) there exists a unique solution \(u_\mu\) of problem \((1')\)—\((4')\), and
\[ \max_{\overline\Pi_{T_2}} |u_\mu|<C<+\infty \]
for \(0<\mu<\mu^*\), where \(C\) is a constant independent of \(\mu\).

Theorem 5. Whatever closed domain \(\Omega\subseteq\overline\Pi_{T_2}\) not having a common point with the characteristic \(t=x+T^*\) may be,
\[ \max_{\Omega}|u_\mu-u_p|+\max_{\Omega}|\partial u_\mu/\partial x-\partial u_p/\partial x|+\max_{\Omega}|\partial u_\mu/\partial t-\partial u_p/\partial t|\to 0 \]
as \(\mu\to 0\).

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

Received
28 II 1968

REFERENCES

\(^1\) V. E. Abolinya, A. D. Myshkis, Matem. sborn., 50 (92), 4, 423 (1960).
\(^2\) V. N. Gol’dberg, Yu. I. Neimark, Matem. sborn., 67 (109), 1, 16 (1965).
\(^3\) V. N. Gol’dberg, DAN, 176, No. 6 (1967).
\(^4\) A. A. Vitt, ZhTF, 6, issue 9 (1936).