UDC 517.944 : 517.946.9

ON PERIODIC SOLUTIONS OF PARABOLIC EQUATIONS

Yu. P. GOR'KOV

Recently, in a number of works, periodic oscillatory processes in certain electrolytic systems and during well blowout have been studied (see [1—4]). It is also known that periodic regimes occur in relay control of the temperature of a body (see [9]). In essence, the mathematical formulation of these problems is the same.

The present work is devoted to the study of the question of the existence of periodic regimes and to the investigation of the behavior of solutions as \(t \to \infty\) for problems on controlling the temperature of a body in the case when the control is carried out by a jump-like change of internal heat sources.

Let us introduce the following notation. Let \(\Omega\) be some bounded domain of \(n\)-dimensional Euclidean space \(E_n(x_1, x_2, \ldots, x_n)\) with sufficiently smooth boundary \(\Gamma\). By \(Q_{(t_1,t_2)}\), \(Q_{(t_1,t_2]}\), \(Q_{[t_1,t_2)}\) we shall denote the \(n+1\)-dimensional cylindrical domains in the space \((x,t)=(x_1,x_2,\ldots,x_n,t)\), consisting of those points for which \(x \in \Omega\) and respectively \(t \in (t_1,t_2)\), \(t \in [t_1,t_2)\). The lateral surface \(Q_{(t_1,t_2)}\) \((x \in \Gamma,\ t_1 < t < t_2)\) will be denoted by \(S_{(t_1,t_2)}\), and the closures of \(Q_{(t_1,t_2)}\) and \(S_{(t_1,t_2)}\) by \(\overline Q_{(t_1,t_2)}\), \(\overline S_{(t_1,t_2)}\). Suppose that for \(x \in \Omega\) the operator is defined
\[ L \equiv \sum_{i,j=1}^{n} a_{ij}(x)\frac{\partial^2}{\partial x_i \partial x_j} + \sum_{i=1}^{n} b_i(x)\frac{\partial}{\partial x_i} + c(x). \]
Assume that \(a_{ij}(x)\), \(b_i(x)\), \(c(x)\) are sufficiently smooth functions in \(\overline\Omega\), \(c(x) \leqslant 0\), and
\[ \sum_{i,j=1}^{n} a_{ij}(x)\xi_i\xi_j \geq a_0 \sum_{j=1}^{n}\xi_j^2 \]
for all real \(\xi_j\) and some constant \(a_0>0\). Finally, we shall suppose that \(\xi\) is some fixed point of the domain \(\Omega\) and \(d_1,\ d_2\) are two numbers satisfying the condition \(d_1>d_2\).

1. Consider in \(Q_{(0,\infty)}\) the following problem:
   \[ \frac{\partial u}{\partial t}-Lu= \begin{cases} f_1(x),\\ f_2(x), \end{cases} \tag{1} \]
   \[ u\big|_{S_{(0,\infty)}}=0, \tag{2} \]
   where \(f_i(x)\) are sufficiently smooth functions in \(\overline\Omega\) satisfying the conditions
   \[ L(f_1-f_2)<0\quad (x\in\Omega), \qquad f_1-f_2=0\quad (x\in\Gamma). \tag{3} \]

By virtue of the maximum principle for an elliptic equation, it follows from condition (3) that

\[ f_1(x) > f_2(x) \quad (x \in \Omega). \tag{4} \]

A function \(u(x,t)\), continuous in \(\overline Q_{(0,\infty)}\), will be called a solution of problem (1), (2) if

1) for every fixed \(T>0\) there exists only a finite number of values of \(t\): \(t_1,t_2,\ldots,t_N\) \((0<t_i<T,\ t_i<t_{i+1},\ i=1,2,\ldots,N-1)\), such that

a) either \(u(\xi,t_{i+1})=d_1\), and then in \(Q_{(t_i,t_{i+1})}\) \(u(x,t)\) is a solution of the equation
\[ \frac{\partial u}{\partial t}-Lu=f_1(x), \]
where
\[ u(\xi,t)<d_1 \quad \text{for } t_i \le t<t_{i+1}, \tag{5} \]

b) or \(u(\xi,t_{i+1})=d_2\), and then in \(Q_{(t_i,t_{i+1})}\) \(u(x,t)\) is a solution of the equation
\[ \frac{\partial u}{\partial t}-Lu=f_2(x), \]
where
\[ u(\xi,t)>d_2 \quad \text{for } t_i \le t<t_{i+1}; \tag{6} \]

2) \(u|_{S_{(0,\infty)}}=0\).

Denote by \(v_i(x)\) the solutions of the Dirichlet problems:
\[ Lv_i=-f_i(x)\quad (x\in\Omega),\qquad v_i(x)|_{\Gamma}=0 \quad (i=1,2). \]

Obviously, \(v_1(x)>v_2(x)\) \((x\in\Omega)\).

Suppose now that

\[ v_2(\xi)<d_2<d_1<v_1(\xi). \tag{7} \]

Theorem 1. Let conditions (3), (7) be satisfied. Then in the domain \(Q_{(0,\infty)}\) there exists a periodic solution of problem (1), (2).

We first establish the validity of several auxiliary propositions used in the proof of Theorem 1.

Lemma 1. Let a continuous function \(u(x,t)\) be defined in the domain \(\overline Q_{(t^*,\infty)}\), satisfying the condition \(u|_{S_{(t^*,\infty)}}=0\). Let there exist a sequence of numbers \(\{t_n\}\) \((t_n<t_{n+1},\ t_0=t^*,\ t_n\to\infty\) as \(n\to\infty)\) and a sequence of functions \(\{f_n(x,t)\}\) such that:

a) in \(Q_{(t_n,t_{n+1}]}\) \(u(x,t)\) is a solution of the equation
\[ \frac{\partial u}{\partial t}-Lu=f_n(x,t), \]

b) \(|f_n(x,t)|\le M_1\) for \((x,t)\in \overline Q_{(t_n,t_{n+1})}\) \((n=0,1,2,\ldots)\). Then the estimate \(|u(x,t)|\le M_2\) holds for \((x,t)\in \overline Q_{(t^*,\infty)}\), where the constant \(M_2\) depends only on the coefficients \(a_{ij}, b_i, c\), the constant \(M_1\), and \(\max_{x\in\overline\Omega}|u(x,t^*)|\).

Proof. Introduce the notation \(N_1=\max_{x\in\overline\Omega}|u(x,t^*)|\). Let \(w(x)\) be the solution of the Dirichlet problem: \(Lw=-M_3\) \((x\in\Omega)\), \(w|_{\Gamma}=N_1\), where \(M_3\) is a certain constant satisfying the conditions: \(M_3-|f_n(x,t)|\ge0\) \(((x,t)\in Q_{(t_n,t_{n+1})},\ n=0,1,2,\ldots)\), \(M_3+c(x)N_1\ge0\) \((x\in\Omega)\). Since \(L(w-N_1)=-M_3-c(x)N_1\le0\) and \(w|_{\Gamma}=N_1\), it follows that \(w(x)\ge N_1\) for \(x\in\Omega\).

Further, from the inequalities

\[ L(w-u)-\frac{\partial}{\partial t}(w-u)=-M_3-f_n(x,t)\leqslant 0 \]

\[ \bigl((x,t)\in Q_{(t_n,t_{n+1})},\quad n=0,1,2,\ldots\bigr); \]

\[ (w-u)\big|_{S_{(t^*,\infty)}}\geqslant 0,\qquad w-u(x,t^*)\geqslant 0 \]

by virtue of the maximum principle, applied to the function \(w-u\) successively in the domains \(Q_{(t^*,t_1)}\), \(Q_{(t_1,t_2)}\), ..., \(Q_{(t_n,t_{n+1})}\), ..., we have \(w(x)\geqslant u(x,t)\) for \((x,t)\in Q_{(t^*,\infty)}\). Similarly it is proved that \(u(x,t)\geqslant -w(x)\) \(\bigl((x,t)\in Q_{(t^*,\infty)}\bigr)\). Lemma 1 is proved.

Lemma 2. There exists a continuous monotonically decreasing function \(\varphi(t)\) \(\bigl(\varphi(0)=1,\ \lim_{t\to\infty}\varphi(t)=0\bigr)\), such that for any function \(u(x,t)\), continuous in the domain \(\overline{Q}_{(t^*,\infty)}\), which is in \(Q_{(t^*,\infty)}\) a solution of the equation

\[ \frac{\partial u}{\partial t}-Lu=0 \tag{8} \]

and satisfying the condition \(u\big|_{S_{(t^*,\infty)}}=0\), the estimate

\[ |u(x,t)|\leqslant \max_{x\in\overline{\Omega}} |u(x,t^*)|\,\varphi(t-t^*) \]

holds.

Proof. Denote by \(v(x,t)\) the solution of the equation

\[ \frac{\partial v}{\partial t} -\sum_{i,j=1}^{n} a_{ij}(x)\frac{\partial^2 v}{\partial x_i\partial x_j} -\sum_{i=1}^{n} b_i(x)\frac{\partial v}{\partial x_i}=0 \tag{9} \]

\[ \bigl((x,t)\in Q_{(0,\infty)}\bigr), \]

satisfying the conditions: \(v(x,0)=1,\ v\big|_{S_{(0,\infty)}}=\psi(t)\), where \(\psi(t)\) is some sufficiently smooth function subject to the conditions:
\(\psi(0)=1,\ \psi'(0)=0,\ \psi'(t)<0\) \((t>0)\), \(\lim_{t\to\infty}\psi(t)=0\). By the properties of the function \(v(x,t)\) (see [5]) the derivative \(v_t\) satisfies equation (9) for \((x,t)\in Q_{(0,\infty)}\) and the conditions: \(v_t\big|_{t=0}=0,\ v_t\big|_{S_{(0,\infty)}}=\psi'(t)<0\). From the strong maximum principle (see, for example, [6], § 1, theorem 6) it follows that \(v_t<0\) for \((x,t)\in Q_{(0,\infty)}\).

Let now \(u(x,t)\) be any function continuous in \(\overline{Q}_{(t^*,\infty)}\), satisfying the condition \(u\big|_{S_{(t^*,\infty)}}=0\) and equation (8) for \((x,t)\in Q_{(t^*,\infty)}\). Put

\[ \tilde u(x,t)=u(x,t-t^*)\qquad \left(N_1=\max_{x\in\overline{\Omega}} |u(x,t^*)|\right). \]

Obviously, the inequalities

\[ \frac{\partial}{\partial t}(vN_1-\tilde u)-L(vN_1-\tilde u)\geqslant 0 \qquad \bigl((x,t)\in Q_{(0,\infty)}\bigr), \]

\[ (vN_1-\tilde u)\big|_{t=0}\geqslant 0,\qquad (vN_1-\tilde u)\big|_{S_{(0,\infty)}}\geqslant 0, \]

hold; from them it follows that \(\tilde u\leqslant vN_1\) \(\bigl((x,t)\in Q_{(0,\infty)}\bigr)\). Similarly the inequality \(\tilde u\geqslant -vN_1\) \(\bigl((x,t)\in Q_{(0,\infty)}\bigr)\) is proved. It is enough to take \(\varphi(t)=\max_{x\in\overline{\Omega}} v(x,t)\). Lemma 2 is proved.

To prove the theorem the following device is used: in the domain \(Q_{(-\infty,\infty)}\) one constructs a function \(z(x,t)\), periodic in \(t\) with period \(\tau_1+\tau_2\), satisfying the equations

\[ \frac{\partial u}{\partial t}-Lu=f_1(x), \tag{10} \]

\[ \frac{\partial u}{\partial t}-Lu=f_2(x) \tag{11} \]

respectively in the domains \(Q_{(i,\tau_1)}\), \(Q_{(\tau_1,\tau_1+\tau_2)}\) (\(\tau_i\) are parameters, with \(\tau_1\geqslant 0,\ \tau_2\geqslant 0,\ \tau_1+\tau_2>0\)). It is then shown that, with a proper choice of \(\tau_1,\tau_2\), all the conditions defining the solution of problem (1), (2) will be satisfied for \(z(x,t)\).

For the equation \(\dfrac{\partial u}{\partial t}-Lu=f(x,t)\) with a smooth right-hand side \(f(x,t)\), periodic in \(t\), a periodic solution was constructed in [7]. Since in our case the right-hand side is discontinuous, the proof of existence of a periodic solution differs somewhat from the proof in [7].

Proof of the theorem. Introduce the notation \(\tau=\tau_1+\tau_2\). Construct a sequence \(\{u_m\}\), where \(u_m\) is defined as a function continuous in the domain \(\overline Q_{(-m\tau,\infty)}\), satisfying the conditions

\[ u_m\big|_{S_{(-m\tau,\infty)}}=0,\qquad u_m(x,-m\tau)=0 \]

and equations (10), (11), respectively, in the domains

\[ Q_{(-j\tau,-j\tau+\tau_1)},\qquad Q_{(-j\tau+\tau_1,-(j-1)\tau)} \quad (j=m,\ m-1,\ m-2,\ldots;\ m\geqslant 0). \]

We shall show that the sequence \(\{u_m\}\) converges as \(m\to\infty\), uniformly in \(x\) and \(t\) from the domain \(\overline Q_{(T_1,\infty)}\) (\(T_1\) is any number), to a certain continuous function \(z(x,t)\). By Lemma 1 we have

\[ |u_m(x,t)|\leqslant M_4\qquad \bigl((x,t)\in Q_{(-m\tau,\infty)}\bigr), \tag{12} \]

where \(M_4\) is a certain constant independent of \(m\). Further, for any natural numbers \(p\) and \(q\) (for definiteness, let \(p>q\)) the function \(u_p-u_q\) is, in the domain \(Q_{(-q\tau,\infty)}\), a solution of equation (8). Applying Lemma 2 to it, we obtain

\[ |u_p(x,t)-u_q(x,t)| \leqslant \max_{x\in\overline\Omega}|u_p(x,-q\tau)-u_q(x,-q\tau)|\,\varphi(t+q\tau) \]

or, by virtue of (12),

\[ |u_p(x,t)-u_q(x,t)|\leqslant 2\cdot M_4\,\varphi(t+q\tau). \]

Since \(\varphi(t)\to 0\) as \(t\to\infty\), the last inequality implies the uniform convergence of the sequence \(\{u_m\}\) as \(m\to\infty\) and \((x,t)\in\overline Q_{(T_1,\infty)}\).

We now show that \(z(x,t)\) is a solution of equations (10), (11), respectively, in the domains

\[ Q_{(j\tau,j\tau+\tau_1)},\qquad Q_{(j\tau+\tau_1,(j+1)\tau)} \quad (j=0,\ \pm1,\ \pm2,\ldots). \]

Let \(k\) be any natural number, and let \(z_k(x,t)\) be a function continuous in the domain \(\overline Q_{(-k\tau,\infty)}\) and satisfying the conditions

\[ z_k\big|_{S_{(-k\tau,\infty)}}=0,\qquad z_k(x,-k\tau)=z(x,-k\tau) \]

and equations (10), (11), respectively, in the domains

\[ Q_{(-j\tau,-j\tau+\tau_1)},\qquad Q_{(-j\tau+\tau_1,-(j-1)\tau)} \quad (j=k,\ k-1,\ k-2,\ldots). \]

By the maximum principle,

\[ |u_m(x,t)-z_k(x,t)| \leqslant \max_{x\in\overline\Omega}|u_m(x,-k\tau)-z_k(x,-k\tau)| \qquad (m>k). \]

From this estimate it follows that \(u_m(x,t)\to z_k(x,t)\) as \(m\to\infty\), uniformly in \(x\) and \(t\) from the domain \(\overline Q_{(-k\tau,\infty)}\). On the other hand, \(u_m(x,t)\to z(x,t)\) as

\(m \to \infty\) uniformly in \(x, t\) \(\bigl((x,t)\in \overline Q_{(-k\tau,\infty)}\bigr)\). Consequently, \(z(x,t)\equiv z_k(x,t)\) for \((x,t)\in \overline Q_{(-k\tau,\infty)}\). It is easy to see that \(z(x,t)\) is periodic in \(t\) with period \(\tau=\tau_1+\tau_2\). Indeed, \(\widetilde z(x,t)=z(x,t+\tau)-z(x,t)\) is a solution of equation (8) for \((x,t)\in Q_{(-\infty,\infty)}\). By Lemma 2 we have

\[ |\widetilde z(x,t)| \le \max_{x\in \overline \Omega} |\widetilde z(x,T_1)|\,\varphi(t-T_1), \qquad \bigl((x,t)\in Q_{(T_1,\infty)}\bigr). \]

Since \(z(x,t)\) is uniformly bounded in \(Q_{(-\infty,\infty)}\) (this follows from estimate (12)), it follows from the last inequality that \(\widetilde z(x,t)\equiv 0\) in \(\overline Q_{(-\infty,\infty)}\).

We now investigate the dependence of the function \(z(x,t)\) on the parameters \(\tau_1,\tau_2\). First of all, let us show that
\[ \sup_{(x,t)\in Q_{(0,T_2)}} |z^*-z|\to 0 \]
as \(\tau_i^*\to \tau_i\) (\(T_2\) is any positive number). We shall prove this assertion for the case when \(\tau_i^*>\tau_i>0\). In the other cases the proof is carried out analogously.

Introduce the notation \(w(x,t)=z^*(x,t)-z(x,t)\). By Lemma 2 and the known estimates for solutions of parabolic equations ([6], § 1, Theorem 2), we have

\[ |w(x,\tau_1)| \le \max_{x\in \overline \Omega} |w(x,0)|\,\varphi(\tau_1), \]

\[ |w(x,\tau_1^*)| \le \max_{x\in \overline \Omega} |w(x,\tau_1)| +(\tau_1^*-\tau_1)\max_{x\in \overline \Omega}(f_1(x)-f_2(x)), \]

\[ |w(x,\tau)| \le \max_{x\in \overline \Omega} |w(x,\tau_1^*)|\,\varphi(\tau-\tau_1^*), \]

\[ |w(x,\tau^*)| \le \max_{x\in \overline \Omega} |w(x,\tau)| +(\tau^*-\tau)\max_{x\in \overline \Omega}(f_1(x)-f_2(x)). \]

On the other hand,

\[ \begin{aligned} \max_{x\in \overline \Omega}|w(x,\tau^*)| &\ge \max_{x\in \overline \Omega}|z^*(x,\tau^*)-z(x,\tau)| -\max_{x\in \overline \Omega}|z(x,\tau^*)\\ &\quad -z(x,\tau)| = \max_{x\in \overline \Omega}|w(x,0)| -\max_{x\in \overline \Omega}|z(x,\tau^*)-z(x,\tau)|. \end{aligned} \]

Comparing the inequalities obtained, we get the relation

\[ \max_{x\in \overline \Omega}|w(x,0)| \le \]

\[ \le \frac{ \max_{x\in \overline \Omega}|z(x,\tau^*)-z(x,\tau)| +(\tau^*-\tau)\cdot \max_{x\in \overline \Omega}(f_1(x)-f_2(x)) }{ 1-\varphi(\tau_1)\varphi(\tau-\tau_1^*) } + \]

\[ + \frac{ (\tau_1^*-\tau_1)\varphi(\tau-\tau_1^*)\max_{x\in \overline \Omega}(f_1(x)-f_2(x)) }{ 1-\varphi(\tau_1)\varphi(\tau-\tau_1^*) }, \]

from which it follows that \(w(x,0)\to 0\) as \(\tau_i^*\to \tau_i\). But then, obviously, \(w(x,t)\to 0\) as \(\tau_i^*\to \tau_i\) \(\bigl((x,t)\in Q_{(0,T_2)}\bigr)\). Note that, as \(\tau_1\to \widetilde\tau>0\), \(\tau_2\to 0\) (as \(\tau_1\to 0\), \(\tau_2\to \widetilde\tau>0\)), \(z(x,t)\to v_1(x)\) \(\bigl(z(x,t)\to v_2(x)\bigr)\) uniformly in \(x,t\) from the domain \(Q_{(-\infty,\infty)}\).

Let now \(z(x,t)\), \(\overline z(x,t)\) be periodic functions corresponding to the parameter values \(\tau_i,\tau_i'\) \((i=1,2)\). Let \(\tau_1+\tau_2=\tau_1'+\tau_2'\), \(\tau_1'>\tau_1\).

We shall show that \(z(x,t)<\bar z(x,t)\) for \((x,t)\in Q_{(-\infty,\infty)}\). Since in \(Q_{(-\infty,\infty)}\)

\[ L(\bar z-z)-\frac{\partial}{\partial t}(\bar z-z)\leq 0, \]

it follows from Remark 1 to Theorem 3 (§ 12) of [6] that

\[ \lim_{t\to\infty}\inf_{x\in\Omega}(\bar z-z)\gg 0. \]

Since \(\bar z-z\) is a function periodic in \(t\), we have \(\bar z-z\gg 0\) for \((x,t)\in Q_{(-\infty,\infty)}\). Further, by virtue of inequality (4) we have \(\bar z(x,t)-z(x,t)>0\) for \((x,t)\in Q_{(\tau_1,\tau'_1)}\). Finally, in the domains \(Q_{(0,\tau_1]}\) (for \(\tau_1>0\)) and \(Q_{(\tau'_1,\tau]}\) (for \(\tau'_1<\tau\)) the strict inequality follows from the strong maximum principle.

We now show that \(z_t(x,t)>0\) for \((x,t)\in Q_{(0,\tau_1]}\), \(z_t(x,t)<0\) for \((x,t)\in Q_{(\tau_1,\tau]}\), and \(|z_t(x,t)|\leq M_5\) for \((x,t)\in \bar Q_{(-\infty,\infty)}\) (\(\tau_i>0\), \(M_5\) is a certain constant independent of \(\tau_i\), \(i=1,2\)). By virtue of the properties of \(z(x,t)\), the derivative \(z_t(x,t)\) in \(Q_{(0,\tau_1]}\), \(Q_{(\tau_1,\tau]}\) satisfies equation (8). Since

\[ \lim_{t\to\tau_1-0} z_t=\lim_{t\to\tau_1+0} z_t+f_1(x)-f_2(x),\qquad \lim_{t\to\tau-0} z_t=\lim_{t\to\tau+0} z_t-f_1(x)+f_2(x) \]

and \(z(x,t)\) is a periodic function (with period \(\tau=\tau_1+\tau_2\)), the function

\[ p(x,t)= \begin{cases} z_t(x,t), & (x,t)\in \bar Q_{(m\tau,\;m\tau+\tau_1)},\\ z_t(x,t)+f_1(x)-f_2(x), & (x,t)\in \bar Q_{(m\tau+\tau_1,\;(m+1)\tau)} \end{cases} \]

is continuous in \(\bar Q_{(-\infty,\infty)}\) and satisfies the equations

\[ Lp-\frac{\partial p}{\partial t}=0, \]

\[ Lp-\frac{\partial p}{\partial t}=L(f_1-f_2)\quad\text{in } Q_{(m\tau,\;m\tau+\tau_1)},\quad Q_{(m\tau+\tau_1,\;(m+1)\tau)}\quad (m=0,\pm1,\pm2,\ldots) \]

respectively. Clearly, \(p(x,t)>0\) for \((x,t)\in Q_{(-\infty,+\infty)}\). But then \(z_t(x,t)>0\) in \(Q_{(0,\tau_1]}\). Similarly it is proved that \(z_t(x,t)<0\) for \((x,t)\in Q_{(\tau_1,\tau]}\). The estimate \(|z_t(x,t)|\leq M_5\) follows from Lemma 1.

Introduce the following notation:

\[ A(\tau_1,\tau_2)=z(\xi,\tau_1),\qquad B(\tau_1,\tau_2)=z(\xi,\tau). \]

The functions \(A\) and \(B\), as was already shown above, depend continuously on \(\tau_1,\tau_2\) and in the domain \(\{\tau_i>0\}\) are monotonically decreasing with respect to \(\tau_2\) when \(\tau_1=\tau_0-\tau_2\) \((0<\tau_0<\infty)\). Moreover, \(A(\tau_1,0)=v_1(\xi)\gg d_1\), \(B(0,\tau_2)=v_2(\xi)<d_2\). Consequently, the sets of points where \(A=d_1\) and \(B=d_2\) are continuous curves, each of which intersects any straight line of the family \(\tau_1+\tau_2=\tau_0\) at one and only one point.

Let \((\alpha(\tau_0),\tau_0-\alpha(\tau_0))\), \((\beta(\tau_0),\tau_0-\beta(\tau_0))\) be the coordinates of the points of intersection of the curves \(\{A=d_1\}\), \(\{B=d_2\}\) with the straight line \(\tau_1+\tau_2=\tau_0\). We shall show that, for sufficiently large values of \(\tau_0\), \(\alpha(\tau_0)<\beta(\tau_0)\). Denote by \(\tilde z_i(x,t)\) the solutions of the equations

\[ \frac{\partial \tilde z_1}{\partial t}-L\tilde z_1=f_2(x),\qquad \frac{\partial \tilde z_2}{\partial t}-L\tilde z_2=f_1(x)\qquad ((x,t)\in Q_{(0,\infty)}), \]

satisfying the conditions \(\tilde z_i\big|_{S_{(0,\infty)}}=0\), \(\tilde z_i(x,0)=v_i(x)\). Since the functions \(\tilde z_1(x,t)\), \(\tilde z_2(x,t)\) tend, as \(t\to\infty\), uniformly in \(x\) to the functions \(v_2(x)\), \(v_1(x)\), respectively (see [6], p. 140, Remark 2), it follows that there exist-

there exist \(\bar t_1,\bar t_2\) such that \(\tilde z_1(\xi,\bar t_1)=d_2\), \(\tilde z_2(\xi,\bar t_2)=d_1\), and moreover \(\tilde z_1(\xi,t)>d_2\) for \(0\le t<\bar t_1\), \(\tilde z_2(\xi,t)<d_1\) for \(0\le t<\bar t_2\). From the inequalities

\[ z(x,t+\tau_1)\le \tilde z_1(x,t)\quad \bigl((x,t)\in Q_{(0,\tau_2)}\bigr), \]

\[ \tilde z_2(x,t)\le z(x,t)\quad \bigl((x,t)\in Q_{(0,\tau_1)}\bigr), \]

which hold by virtue of the maximum principle, there follow the inequalities

\[ \alpha(\tau_0)<\bar t_2,\qquad \tau_0-\beta(\tau_0)<\bar t_1. \]

Since \(\alpha(\tau_0)\), \(\beta(\tau_0)\) are defined for all \(\tau_0>0\), it follows from the last inequalities that \(\alpha(\tau_0)<\beta(\tau_0)\) for \(\tau_0>\bar t_1+\bar t_2\). On the other hand, for sufficiently small values of \(\tau_0\) the opposite inequality holds,

\[ \alpha(\tau_0)>\beta(\tau_0). \]

Indeed, otherwise there would have to exist a sequence of points \(\{(\tau_{1n},\tau_{2n})\}\) \((\tau_{in}>0)\) such that \(A(\tau_{1n},\tau_{2n})\ge d_1\), \(B(\tau_{1n},\tau_{2n})\le d_2\), and \(\lim_{n\to\infty}(\tau_{1n}+\tau_{2n})=0\). In view of the uniform boundedness of \(z_t\), this cannot occur. Thus it has been proved that the system of equations

\[ A(\tau_1,\tau_2)=d_1,\qquad B(\tau_1,\tau_2)=d_2 \tag{13} \]

has a solution.

The periodic function \(z(x,t)\) corresponding to the solution of system (13) will evidently be a solution of problem (1), (2). The theorem is proved.

Remark. If, in the definition of a solution of problem (1), (2), one abandons the requirement that inequalities (5), (6) be satisfied, then, as is clear from the proof of the theorem, a periodic solution exists under weaker restrictions on the functions \(f_i(x)\), namely under the condition that \(f_1(x)>f_2(x)\) \((x\in\Omega)\).

1. We now investigate the behavior, as \(t\to\infty\), of the solution of the following problem:

\[ \frac{\partial u}{\partial t} -\sum_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^2 u}{\partial x_i\partial x_j} -\sum_{i=1}^{n} b_i(x,t,u)\frac{\partial u}{\partial x_i} = \begin{cases} \varphi_1(u),\\ \varphi_2(u), \end{cases} \tag{14} \]

\[ \left. \frac{\partial u}{\partial \nu}\equiv \sum_{i,j=1}^{n} a_{ij}(x,t)\cos(\mathbf n,x_i)\frac{\partial u}{\partial x_j} \right|_{S_{(0,\infty)}}=0, \tag{15} \]

\[ u(x,0)=u_0(x), \tag{16} \]

where \(\mathbf n\) is the inward normal to \(\Gamma\).

Suppose the following conditions are satisfied:

\[ 1^\circ.\quad \sum_{i,j=1}^{n} a_{ij}(x,t)\xi_i\xi_j\ge a_1\sum_{i=1}^{n}\xi_i^2 \]

for all real \(\xi_i\) and some constant \(a_1>0\), \(\bigl((x,t)\in \overline Q_{(0,\infty)}\bigr)\);

\[ 2^\circ.\quad 0<\varphi_1(u),\qquad \varphi_1'(u)<0,\qquad \varphi_2(u)<0,\qquad \varphi_2'(u)<0 \quad (|u|<\infty). \]

In addition, let \(a_{ij}(x,t)\), \(b_i(x,t,u)\), \(\varphi_i(u)\), \(u_0(x)\) be sufficiently smooth functions for \(x\in\Omega\), \(|u|<\infty\), \(t\ge0\).

Denote

\[ \tilde L u\equiv \sum_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^2 u}{\partial x_i\partial x_j} + \sum_{i=1}^{n} b_i(x,t,u)\frac{\partial u}{\partial x_i}. \]

By a solution of problem (14)—(16) we shall mean a function \(u(x,t)\), continuous together with the derivatives \(\dfrac{\partial u}{\partial x_i}\) in the domain \(\overline Q_{(0,\infty)}\), satisfying conditions (15), (16) and condition 1), in which the equations \(\dfrac{\partial u}{\partial t}-Lu=f_i(x)\) are replaced respectively by the equations \(\dfrac{\partial u}{\partial t}-\widetilde L u=\varphi_i(u)\) \((i=1,2)\).

If \(u_0(x)\equiv d_2\), then it is easy to see that the solution of problem (14)—(16) will depend only on \(t\), and will be periodic with some period \(\tau\). Denote the solution in this case by \(z(t)\). Suppose now that the following conditions are satisfied:

\[ d_2<u_0(x)<d_1\quad (x\in \overline\Omega),\qquad \left.\frac{\partial u_0}{\partial \nu}\right|_{\Gamma}=0 . \tag{17} \]

We shall first show that problem (14)—(16) has a solution. Let \(\{t_n\}\) be some sequence of numbers satisfying the conditions \(t_0=0\), \(t_0<t_1<t_2<\cdots<t_n<t_{n+1}<\cdots\), \(t_n\to\infty\) as \(n\to\infty\). Denote by \(\bar u(x,t)\) a function continuous in the domain \(\overline Q_{(0,\infty)}\), satisfying conditions (15), (16) and the equations

\[ \frac{\partial u}{\partial t}-\widetilde L u=\varphi_1(u), \tag{18} \]

\[ \frac{\partial u}{\partial t}-\widetilde L u=\varphi_2(u) \tag{19} \]

respectively in the domains \(Q_{(t_{2m},\,t_{2m+1})}\), \(Q_{(t_{2m+1},\,t_{2m+2})}\) \((m=0,1,2,\ldots)\). The existence of such a function follows from Theorem 8 of [8].

Next, denote by \(V_1^n(t)\), \(V_2^n(t)\) the solutions of the Cauchy problem for the equation \(\dfrac{dV}{dt}=\varphi_{a_n}(V)\) \((t\ge t_n)\) with initial conditions

\[ V_1^n(t_n)=\max_{x\in\overline\Omega}\bar u(x,t_n),\qquad V_2^n(t_n)=\min_{x\in\overline\Omega}\bar u(x,t_n) \]

\[ \left( a_n= \begin{cases} 1 & \text{for } n \text{ even},\\ 2 & \text{for } n \text{ odd}, \end{cases} \quad n=0,1,2,\ldots \right). \]

Using condition \(2^\circ\), it is easy to verify that

\[ (-1)^n V_i^n(t)\to\infty \quad (\text{as } t\to\infty). \tag{20} \]

The functions \(\bar u(x,t)-V_i^n(t)\) in the domain \(Q_{(t_n,t_{n+1})}\) satisfy respectively the equations

\[ \frac{\partial}{\partial t}\left(\bar u-V_i^n\right) -\sum_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^2}{\partial x_i\partial x_j} \left(\bar u-V_i^n\right)- \]

\[ -\sum_{i=1}^{n} b_i(x,t,u)\frac{\partial}{\partial x_i} \left(\bar u-V_i^n\right)= \]

\[ = (\overline u - V_i^n)\int_0^1 \varphi'_{\alpha_n}\bigl(V_i^n+\tau(\overline u-V_i^n)\bigr)\,d\tau \quad (i=1,2) \]

and the relations

\[ \overline u(x,t_n)-V_1^n(t_n)\leq 0,\qquad \left.\frac{\partial}{\partial \nu}(\overline u-V_1^n)\right|_{S(t_n,t_{n+1})}=0, \]

\[ \overline u(x,t_n)-V_2^n(t_n)\geq 0,\qquad \left.\frac{\partial}{\partial \nu}(\overline u-V_2^n)\right|_{S(t_n,t_{n+1})}=0. \]

By virtue of the maximum principle we have

\[ V_2^n(t)\leq \overline u(x,t)\leq V_1^n(t)\quad \text{for } (x,t)\in Q_{(t_n,t_{n+1})}\quad (n=0,1,2,\ldots). \tag{21} \]

On the basis of conditions (20), (21), the sequence \(\{t_n\}\) can be chosen so that

\[ \overline u(\xi,t_{2m+1})=d_1,\qquad \overline u(\xi,t)<d_1 \quad \text{for } t_{2m}\leq t<t_{2m+1}, \]

\[ \overline u(\xi,t_{2m+2})=d_2,\qquad \overline u(\xi,t)>d_2 \quad \text{for } t_{2m+1}\leq t<t_{2m+2}\quad (m=0,1,2,\ldots). \]

Let us show that, with such a choice of the sequence, \(t_n\to\infty\) as \(n\to\infty\), and consequently \(\overline u(x,t)\) is a solution of problem (14)—(16). Moreover, we shall show that \(t_{n+1}-t_n\geq \gamma>0\) (\(\gamma\) is a certain constant, \(n=0,1,2,\ldots\)). The function \(V_1^n-V_2^n\) is a solution of the equation

\[ \frac{d}{dt}(V_1^n-V_2^n) =(V_1^n-V_2^n)\int_0^1 \varphi'_{\alpha_n}\bigl(V_2^n+\tau(V_1^n-V_2^n)\bigr)\,d\tau, \]

or, what is the same thing, satisfies the equality

\[ V_1^n-V_2^n = \left.(V_1^n-V_2^n)\right|_{t=t_n} e^{\int_{t_n}^{t} dt \int_0^1 \varphi'_{\alpha_n}\,d\tau} \quad (t\geq t_n). \tag{22} \]

Since \(\varphi'_{\alpha_n}<0\) and \(V_2^n\leq d_1\) for \(t_n\leq t\leq t_{n+1}\), it follows from equality (22) that

\[ |V_i^n|\leq M_6 \quad (t_n\leq t\leq t_{n+1}), \]

\[ V_1^n-V_2^n \leq \left.(V_1^n-V_2^n)\right|_{t=t_n} e^{-\mu(t-t_n)}, \tag{23} \]

where \(M_6\) is a certain constant independent of \(n\), and

\[ \mu=\min\left\{\min_{|\eta|\leq 2M_6}|\varphi_1'(\eta)|,\ \min_{|\eta|\leq 2M_6}|\varphi_2'(\eta)|\right\} \quad (n=0,1,2,\ldots). \]

Denote by \(\tau_1,\tau_2\) the values of \(t\) such that \(V_1^1(\tau_1)=d_1\), \(\overline V(\tau_2)=d_2\) (here \(\overline V(t)\) is the solution of the equation \(\dfrac{dV}{dt}=\varphi_2(V)\) with the initial condition

\[ \overline V(0)=\min_{x\in\overline\Omega} u_0(x). \]

Since \(V_1^n(t_n)-V_2^n(t_n)\geq V_1^{n+1}(t_{n+1})-V_2^{n+1}(t_{n+1})\) and

\[ V_1^{2m}(t_{2m+1})\geq d_1,\qquad V_2^{2m+1}(t_{2m+2})\leq d_2, \]

we have \(t_{2m+1}-t_{2m}\geq \tau_1,\quad t_{2m+2}-t_{2m+1}\geq \tau_2\)

\[ (n,m=0,1,2,\ldots). \]

Theorem 2. Suppose that conditions \(1^\circ\), \(2^\circ\), (17) are satisfied. Then there exists a constant \(\tau_1\), depending on the solution \(u(x,t)\) of problem (14)—(16), such that
\[ \max_{x\in\overline\Omega}|u(x,t+\tau_1)-z(t)|\to 0 \quad \text{as } t\to\infty. \]

Proof. Without loss of generality, we shall assume that \(u(x,t)\) satisfies equation (18) in the domain \(Q_{(0,t_1)}\). Let \(\{t_n^*\}\) be a sequence of numbers satisfying the conditions

\[ t_0^* = 0,\qquad z(t_{2m+1}^*)=d_1,\qquad t_{2m+2}^* - t_{2m}^* = \tau \quad (m=0,1,2,\ldots). \]

Denote
\[ \Delta t(n)=(t_n-t_{n-1})-(t_n^*-t_{n-1}^*) \quad (n=1,2,\ldots). \]
We shall show that the series

\[ \sum_{n=1}^{\infty} |\Delta t(n)| < \infty . \tag{24} \]

Let \(t'_{2m}\), \(t''_{2m}\) be values of \(t\) such that
\[ V_1^{2m-1}(t'_{2m})=V_2^{2m-1}(t''_{2m})=d_2 . \]
Taking into account the convexity of \(V_i^{2m-1}\) (indeed,
\[ \frac{d^2}{dt^2} V_i^{2m-1}=\varphi_2'\varphi_2>0 \])
and inequality (23), we have

\[ t'_{2m}-t''_{2m} \leq \frac{V_1^{2m-1}(t'_{2m})-V_2^{2m-1}(t'_{2m})} {\left|\varphi_2'\!\left(V_2^{2m-1}(t'_{2m})\right)\right|} \leq \frac{V_1^{2m-1}(t_{2m})-V_2^{2m-1}(t_{2m})} {\left|\varphi_2'\!\left(V_2^{2m-1}(t_{2m})\right)\right|} \leq \]

\[ \leq (d_1-d_2)e^{-\mu t_{2m}} \frac{1}{\left|\varphi_2'\!\left(V_2^{2m-1}(t_{2m})\right)\right|} \leq M_7 e^{-\mu t_{2m}}, \]

where \(M_7\) is a constant independent of \(m\). Since
\[ t''_{2m}\leq t_{2m-1}+(t^*_{2m}-t^*_{2m-1})\leq t'_{2m}, \]
it follows that
\[ |\Delta t(2m)|\leq t'_{2m}-t''_{2m}. \]
Consequently,
\[ |\Delta t(2m)|\leq M_7 e^{-\mu t_{2m}}. \]
Analogously one proves that
\[ \Delta t(2m+1)\leq M_8 e^{-\mu t_{2m+1}}. \]

From the convergence of the series (24) it follows that the sequence \(\{t_n-t_n^*\}\) converges to some constant as \(n\to\infty\). Obviously, this constant must be taken as \(\tau_1\). Theorem 2 is proved.

In conclusion I express my deep gratitude to A. M. Il’in for posing the problem and for his attentive guidance of the work.

References

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Received by the editors
August 5, 1965

Sverdlovsk Branch
of the V. A. Steklov
Mathematical Institute