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Yu. S. Kolesov
On Some Criteria for the Existence of Stable Periodic Solutions of Quasilinear Parabolic Equations
(Presented by Academician I. N. Vekua on 30 III 1964)
1. Let \(\Omega\) be a bounded open domain in \(n\)-dimensional space whose boundary \(\Gamma\) is sufficiently smooth. Consider the parabolic equation
\[ \frac{\partial u}{\partial t}=Lu+f(t,x,u). \tag{1} \]
Here \(L\) is a second-order elliptic operator:
\[ Lu=\sum_{i,k=1}^{n} a_{ik}(x)\frac{\partial^2 u}{\partial x_i\partial x_k} +\sum_{k=1}^{n} a_k(x)\frac{\partial u}{\partial x_k} -a(x)u, \tag{2} \]
where
\[ \sum_{i,k=1}^{n} a_{ik}(x)\xi_i\xi_k \ge r_0 \sum_{k=1}^{n}\xi_k^2 \quad (r_0>0), \]
\[ a_{ik}(x)=a_{ki}(x), \qquad a(x)\ge 0. \]
The coefficients of the differential expression (2) are assumed to be sufficiently smooth. The function \(f(t,x,u)=f(t,x_1,\ldots,x_n,u)\) is also assumed to be sufficiently smooth.
We shall assume that \(f(t,x,u)\) has the property of \(\omega\)-periodicity in \(t\): \(f(t+\omega,x,u)\equiv f(t,x,u)\), and we shall consider the question of the existence for equation (1) of solutions \(u(t,x)\) that are \(\omega\)-periodic in \(t\) \((-\infty<t<\infty,\ x\in\overline{\Omega},\ \overline{\Omega}=\Omega+\Gamma)\), satisfying the boundary condition
\[ u(t,x)=0 \quad (x\in\Gamma). \tag{3} \]
To study this problem we shall use the general method proposed by M. A. Krasnosel’skii \({}^{1}\). It consists in writing equation (1), together with the boundary condition (3), in the form of an operator differential equation
\[ \frac{du}{dt}=Lu+f(t,u) \tag{4} \]
in a suitably chosen function space \(E\) (in the theorems proved below, equation (4) is considered in the space \(C_0\) of functions continuous on \(\overline{\Omega}\) and vanishing on \(\Gamma\)). To each initial condition \(u(0)=u_0\) of equation (4) there corresponds the value \(Tu_0\) of the solution of equation (4) at \(t=\omega\). If the operator \(T\) leaves invariant some cone in the space \(E\), then general principles of the existence of fixed points for positive operators are applicable to proving the existence of periodic solutions. If, moreover, the operator \(T\) turns out to be concave, then the corresponding periodic solutions are stable in the sense of Lyapunov. If, on the other hand, the operator \(T\) is convex, then the corresponding periodic solutions are unstable.
The scheme described above was applied to the study of systems of ordinary differential equations in a paper by the author and M. A. Krasnosel’skii—
skii’s paper \((^2)\) and in the paper of M. A. Krasnosel’skii \((^3)\). Some applications to problems on periodic solutions of equations with partial derivatives are indicated in the report of M. A. Krasnosel’skii and P. E. Sobolevskii at the Soviet-American symposium of 1963 in Novosibirsk. Let us also note that some existence theorems for periodic solutions of equations (1) were obtained by D. Kh. Karimov \((^4)\), J. Prodi \((^{5,6})\), and I. I. Shmulev \((^7)\) by other methods.
The transition from problem (1)—(3) to equation (4) and the investigation of the operator \(T\) can be carried out either by classical methods or by methods using ideas from the theory of semigroups \((^{8-10})\).
- All further constructions are carried out under the assumption that
\[ f(t,x,0)\geq 0 \qquad (0\leq t\leq \omega,\ x\in \overline{\Omega}). \tag{5} \]
We denote by \(\lambda_0\) the least eigenvalue of the operator \(Au=-Lu\) \((u(x)=0,\ x\in \Gamma)\) (\(\lambda_0>0\), since this operator has (see (1)) a positive inverse).
Theorem 1. Suppose that the inequality
\[ f(t,x,u)\leq au+a_1 \qquad (0\leq t\leq \omega,\ x\in \overline{\Omega},\ u\geq 0), \tag{6} \]
is satisfied, where \(a<\lambda_0\). Then problem (1)—(3) has at least one nonnegative \(\omega\)-periodic solution.
- In the remaining part of the paper we shall assume that problem (1)—(3) has the zero solution:
\[ f(t,x,0)\equiv 0 \qquad (0\leq t\leq \omega,\ x\in \overline{\Omega}). \tag{7} \]
Theorem 2. Suppose that the conditions of Theorem 1 are satisfied and that
\[ f(t,x,u)\geq bu \qquad (0\leq t\leq \omega,\ x\in \overline{\Omega},\ 0\leq u\leq \rho_0), \tag{8} \]
where \(b>\lambda_0\). Then problem (1)—(3) has at least one \(\omega\)-periodic nonnegative solution distinct from the identically zero solution.
Theorem 3. Suppose that the inequalities
\[ f(t,x,u)> \alpha u-\alpha_1(0\leq t\leq \omega,\ x\in \overline{\Omega},\ u\geq 0), \tag{9} \]
\[ f(t,x,u)\leq \beta u(0\leq t\leq \omega,\ x\in \overline{\Omega},\ 0\leq u\leq \rho_0), \tag{10} \]
are satisfied, where \(\beta<\lambda_0<\alpha\). Then problem (1)—(3) has at least one nonnegative \(\omega\)-periodic solution distinct from the identically zero solution.
- We shall say that the function \(f(t,x,u)\) is strongly concave if
\[ u f'_u(t,x,u)-f(t,x,u)\leq 0 \qquad (0\leq t\leq \omega,\ x\in \overline{\Omega},\ u\geq 0), \tag{11} \]
and the left-hand side is strictly negative for some \(t=t_0\) and all \(x\in \overline{\Omega}\), \(u>0\). It turns out that the strong concavity of the function \(f(t,x,u)\) implies the \(u_0\)-concavity (see \((^1)\)) of the operator \(T\) on the cone \(K\subset C_0\) of functions nonnegative on \(\Omega\).
Theorem 4. Suppose that \(f(t,x,u)\) is strongly concave. Suppose that the conditions of one of Theorems 1 or 2 are satisfied. Then problem (1)—(3) has one and only one nonnegative \(\omega\)-periodic solution distinct from the identically zero solution.
- We shall say that a nonnegative \(\omega\)-periodic solution \(u_0(t,x)\) of problem (1)—(3) is stable if every solution \(u(t,x)\) satisfying a nonzero and nonnegative initial condition has the property that
\[ \lim_{t\to\infty}\|u(t,x)-u_0(t,x)\|=0. \tag{12} \]
Theorem 5. Suppose that the conditions of Theorem 4 are satisfied. Then the nonnegative \(\omega\)-periodic solution of problem (1)—(3), distinct from the identically zero solution, is stable.
Theorem 6. Suppose that the conditions of Theorem 3 are satisfied and that \(f(t,x,u)\) is strongly convex in the sense that
\[ u f'_u(t,x,u)-f(t,x,u)\geqslant 0 \qquad (0\leqslant t\leqslant \omega,\ x\in \overline{\Omega},\ u\geqslant 0) \tag{13} \]
and the left-hand side, for some \(t=t_0\) and all \(x\in \overline{\Omega}\), \(u>0\), is strictly positive. Then every nonnegative and non-identically-zero \(\omega\)-periodic solution of problem (1)—(3) is unstable.
- We note here one general formula that was used in proving the positivity of the operator \(T\) and that, it seems to us, is of independent interest. Let \(u(x)\) \((x\in \overline{\Omega})\) be a continuous function, and let \(\Omega_+\) be the set of points at which \(u(y)=\|u\|=\max_{y\in\Omega}|u(y)|\), and \(\Omega_-\) the set of points at which \(u(y)=-\|u\|\). The following equality holds:
\[ \lim_{\tau\to +0}\frac{1}{\tau}\bigl[\|u(x)+\tau h(x)\|-\|u(x)\|\bigr] = \max\left\{\max_{x\in\overline{\Omega}_+} h(x),\ -\min_{x\in\overline{\Omega}_-} h(x)\right\}. \tag{14} \]
This formula was pointed out by M. A. Krasnosel’skii.
The author expresses his gratitude to M. A. Krasnosel’skii for his guidance.
Received
24 III 1964
CITED LITERATURE
\(^{1}\) M. A. Krasnosel’skii, Positive Solutions of Operator Equations, 1962.
\(^{2}\) Yu. S. Kolesov, M. A. Krasnosel’skii, DAN, 145, No. 6 (1962).
\(^{3}\) M. A. Krasnosel’skii, DAN, 150, No. 3 (1963).
\(^{4}\) D. Kh. Karimov, Tr. Inst. Matem. i Mekh. AN UzSSR, No. 6 (1950).
\(^{5}\) J. Prodi, Rend. Seminar Mat. Univ. Padova, 23, No. 1 (1954).
\(^{6}\) J. Prodi, Atti IV Congr. Unione Mat. Ital., 2, 193, 1953.
\(^{7}\) I. I. Shmulev, DAN, 139, No. 6 (1961).
\(^{8}\) K. Miranda, Equations with Partial Derivatives of Elliptic Type, Moscow, 1957.
\(^{9}\) E. Hopf, Functional Analysis and Semigroups, Moscow, 1951.
\(^{10}\) P. E. Sobolevskii, Tr. Moscow Math. Soc., 10, 297 (1961).