Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Volume 170, No. 5
MATHEMATICS
Yu. S. KOLESOV
ON ONE CRITERION FOR THE EXISTENCE OF PERIODIC SOLUTIONS OF PARABOLIC EQUATIONS
(Presented by Academician I. G. Petrovsky on 10 I 1966)
- Let \(\Omega\) be a bounded open domain of the \(n\)-dimensional space \(E_n\) of points \(x=\{x_1,\ldots,x_n\}\), belonging to the class \(A^{(2,\lambda)}\) (see \((^1)\)). In the domain \(\Omega\) consider the quasilinear parabolic equation
\[ \frac{\partial u}{\partial t} - \sum_{i,k=1}^{n} a_{ik}(t,x,u)\, \frac{\partial^2 u}{\partial x_i \partial x_k} = f\left(t,x,u,\frac{\partial u}{\partial x_1},\ldots,\frac{\partial u}{\partial x_n}\right), \tag{1} \]
where \(a_{ik}(t,x,u)=a_{ki}(t,x,u)\),
\[ \sum_{i,k=1}^{n} a_{ik}(t,x,u)\xi_i\xi_k \ge \gamma(R)\sum_{i=1}^{n}\xi_i^2 \qquad (\gamma(R)>0) \]
for \(-\infty<t<\infty,\ x\in\overline{\Omega}\) \((\overline{\Omega}=\Omega+\Gamma;\ \Gamma\) is the boundary of the domain \(\Omega\)), \(|u|\ge R\) (\(R\) is an arbitrary nonnegative number).
We shall assume that the functions \(a_{ik}(t,x,u)\) and \(f(t,x,u,v_1,\ldots,v_n)\) are \(\omega\)-periodic in \(t\), satisfy a local Hölder condition in the variables \(t\) and \(x\), and, in the variables \(u,v_1,\ldots,v_n\), a local Lipschitz condition.
Finally, we shall assume that
\[ |f(t,x,u,v_1,\ldots,v_n)| \le C_1(R)+C_2(R)\sum_{i=1}^{n}|v_i|^{\,2-\varepsilon_i(R)}, \]
where \(0<\varepsilon_i(R)\le 2,\ -\infty<t<\infty,\ x\in\overline{\Omega},\ |u|\le R\).
We shall be interested in the question of the existence for equation (1) of \(\omega\)-periodic in \(t\) solutions \(u(t,x)\) \((-\infty<t<\infty,\ x\in\Omega)\), satisfying the boundary condition
\[ u(t,x)=0 \qquad (x\in\Gamma). \tag{2} \]
To study this problem we shall use a general method (see \((^2)\)), which consists in the fact that equation (1), together with the boundary condition (2), is written in the form of an operator equation
\[ du/dt+\mathscr{L}(t,u)u=f(t,u) \tag{3} \]
in a suitably chosen functional space \(E\) (in the theorem proved below, equation (3) is considered in the space \(C_0\) of functions continuous on \(\overline{\Omega}\) and vanishing on \(\Gamma\)), then to each initial condition \(u(0)=u_0\) there is assigned the value \(Tu_0\) of the solution of equation (3) at \(t=\omega\). It turns out that the operator \(U=\mathscr{L}^{\alpha}(0,0)T\mathscr{L}^{-\alpha}(0,0)\) is completely continuous in the space \(C_0\) for some \(\alpha\in(1/2,1)\). Clearly, each fixed point \(u_0\) of the operator \(U\) determines the initial value \(\mathscr{L}^{-\alpha}(0,0)u_0\) of an \(\omega\)-periodic solution of equation (3).
- Let there exist functions $\psi_1(t,x)$ and $\psi_2(t,x)$, smooth in the closed domain $\overline{\Omega}$, such that for $0 \leq t \leq \omega$, $x \in \overline{\Omega}$ the inequalities
\[ \frac{\partial \psi_1}{\partial t} - \sum_{i,k=1}^{n} a_{ik}(t,x,\psi_1)\frac{\partial^2 \psi_1}{\partial x_i \partial x_k} \geq f\left(t,x,\psi_1,\frac{\partial \psi_1}{\partial x_1},\ldots,\frac{\partial \psi_1}{\partial x_n}\right), \tag{4} \]
\[ \psi_1(0,x) \geq \psi_1(\omega,x), \tag{5} \]
\[ \frac{\partial \psi_2}{\partial t} - \sum_{i,k=1}^{n} a_{ik}(t,x,\psi_2)\frac{\partial^2 \psi_2}{\partial x_i \partial x_k} \leq f\left(t,x,\psi_2,\frac{\partial \psi_2}{\partial x_1},\ldots,\frac{\partial \psi_2}{\partial x_n}\right), \tag{6} \]
\[ \psi_2(0,x) \leq \psi_2(\omega,x), \tag{7} \]
and for $0 \leq t \leq \omega$, $x \in \Gamma$
\[ \psi_1(t,x) \geq 0,\qquad \psi_2(t,x) \leq 0. \tag{8} \]
We shall assume that for $t=0$, $x \in \Omega$ the functions $\psi_1(t,x)$ and $\psi_2(t,x)$ are related by
\[ \psi_1(0,x) \geq \psi_2(0,x). \tag{9} \]
Theorem. Let there exist functions $\psi_1(t,x)$ and $\psi_2(t,x)$ satisfying conditions (4)—(9). Then equation (1) has at least one $\omega$-periodic solution $u^*(t,x)$ which satisfies the boundary condition (2) and the inequalities
\[ \psi_2(t,x) \leq u^*(t,x) \leq \psi_1(t,x) \qquad (0 \leq t \leq \omega,\ x \in \Omega). \]
If the solutions of equation (3) are nonlocally extendable, then for the proof one must consider the set $S_1$ of elements $u_0 \in C_0$ satisfying the inequalities
\[ \psi_2(0,x) \leq \mathcal{L}^{-\alpha}(0,0)u_0 \leq \psi_1(0,x). \]
This set is convex and closed. From the theorem on differential inequalities (see, for example, (3)) it follows that
\[ \psi_2(0,x) \leq T\mathcal{L}^{-\alpha}(0,0)u_0 \leq \psi_1(0,x). \]
The last inequalities are equivalent to the inequalities
\[ \psi_2(0,x) \leq \mathcal{L}^{-\alpha}(0,0)Uu_0 \leq \psi_1(0,x). \]
Consequently, $Uu_0 \in S_1$ for $u_0 \in S_1$. It can be shown (using the method developed in (4)) that the set $S_2 = US_1$ is compact in the space $C_0$. Therefore the assertion of the theorem follows from the Schauder principle.
The proof is somewhat complicated if the solutions of equation (3) can, in finite time, “go to infinity.”
- Corollary 1. Let
\[ f(t,x,R_1,0,\ldots,0) \leq 0,\qquad f(t,x,-R_2,0,\ldots,0) \geq 0, \]
where $0 \leq t \leq \omega$; $x \in \Omega$; $R_1, R_2$ are some nonnegative numbers. Then equation (1) has at least one $\omega$-periodic solution $u^*(t,x)$ satisfying the inequalities
\[ -R_2 \leq u^*(t,x) \leq R_1. \]
Corollary 2. Suppose that the operator $\mathcal{L}(t,u)$ does not depend on $t$. Let
\[ \overline{f}(x,u,v_1,\ldots,v_n) \geq \max_{0 \leq t \leq \omega} f(t,x,u,v_1,\ldots,v_n), \]
\[ \underline{f}(x,u,v_1,\ldots,v_n) \leq \min_{0 \leq t \leq \omega} f(t,x,u,v_1,\ldots,v_n). \]
Assume that the equations
\[ \mathcal L(u)u=\overline f\left(x,u,\frac{\partial u}{\partial x_1},\ldots,\frac{\partial u}{\partial x_n}\right), \]
\[ \mathcal L(u)u=\underline f\left(x,u,\frac{\partial u}{\partial x_1},\ldots,\frac{\partial u}{\partial x_n}\right) \]
have such solutions \(\overline u(x)\), \(\underline u(x)\) that
\[ \overline u(x)\geqslant \underline u(x)\qquad (x\in\Omega); \]
\[ \overline u(x)\geqslant 0,\qquad \underline u(x)\leqslant 0\qquad (x\in\Gamma). \]
Then equation (1) has at least one \(\omega\)-periodic solution \(u^*(t,x)\), satisfying the boundary condition (2) and the inequalities
\[ \underline u(x)\leqslant u^*(t,x)\leqslant \overline u(x). \]
Corollaries 1 and 2 contain, as special cases, existence theorems for periodic solutions of equation (1) obtained by I. I. Shmulev \((^5)\) and G. Prodi \((^6)\) by other methods.
Received
4 I 1966
CITED LITERATURE
\(^1\) C. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.
\(^2\) M. A. Krasnosel’skii, P. E. Sobolevskii, Materials for the Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963, p. 3.
\(^3\) S. Kaplan, Comm. Pure and Appl. Math., 16, No. 3, 305 (1963).
\(^4\) P. E. Sobolevskii, Tr. Moscow Math. Soc., 10, 298 (1961).
\(^5\) I. I. Shmulev, DAN, 141, No. 6, 1313 (1961).
\(^6\) G. Prodi, Atti IV Congr. Unione mat. ital., 2, 1953, p. 193.