Mem. Differential Equations Math. Phys. 10(1997), 119–121

T. Kiguradze

ON SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS

OF PERIODIC IN A STRIP SOLUTIONS OF NONLINEAR HYPERBOLIC EQUATIONS

(Reported on March 25 and April 8, 1996)

Let b > 0 and D_{b} = ℝ × [0, b]. In the strip D_{b}, consider the hyperbolic equation

\frac{∂^{2}u}{∂x∂y} = f(x, y, u, \frac{∂u}{∂y}),  (1)

where f : D_{b} × ℝ^{2} → ℝ satisfies the local Carathéodory conditions, i.e. f(·, ·, z_{0}, z_{1}) : D_{b} → ℝ is measurable for every (z_{0}, z_{1}) ∈ ℝ^{2}, f(x, y, ·, ·) : ℝ^{2} → ℝ is continuous for almost every (x, y) ∈ D_{b} and the function

sup{|f(·, ·, z_{0}, z_{1})| : |z_{0}| + |z_{1}| ≤ ρ}

is summable on the rectangle [−a, a] × [0, b] for any ρ > 0 a > 0.

By solution of the equation (1) we understand a locally absolutely continuous function u : D_{b} → ℝ (see [2]) satisfying the equation (1) almost everywhere in D_{b}.

We study the case where f is ω-periodic in the first argument for some ω > 0, i.e.,

f(x + ω, y, z_{0}, z_{1}) = f(x, y, z_{0}, z_{1}) for (x, y) ∈ D_{b}, (z_{0}, z_{1}) ∈ ℝ^{2}.

Let φ : ℝ → ℝ be a locally absolutely continuous ω-periodic function. Below we formulate sufficient conditions of existence and uniqueness of the solution of the equation (1) satisfying

u(x, 0) = φ(x), u(x + ω, y) = u(x, y) for (x, y) ∈ D_{b}.  (2)

Note that earlier the problem (1),(2) has been investigated when f is linear or quasilinear with respect to the last two arguments (see. [1,4-10]).

Theorem 1. Let the inequalities

|f(x, y, z_{0}, 0)| ≤ p_{0}(x, y)(1 + |z_{0}|),  (3)
σ(y)[f(x, y, z_{0}, z_{1}) − f(x, y, z_{0}, \overline{z_{1}})] sign(z_{1} − \overline{z_{1}}) ≤ p(x, y)|z_{1} − \overline{z_{1}}|  (4)

take place on D_{b} × ℝ^{2}, where p_{0} : [0, ω] × [0, b] → ℝ_{+} is summable, σ : [0, b] → {−1, 1} is measurable and p : [0, ω]×[0, b] → ℝ is a measurable function such that p(x, ·) : [0, b] → ℝ is continuous almost for every x, max{|p(·, t)| : t ∈ [0, b]} : [0, ω] → ℝ_{+} is summable and

∫_{0}^{ω} p(s, y) ds < 0 for y ∈ [0, b].  (5)

Then the problem (1), (2) is solvable.

1991 Mathematics Subject Classification. 35L55.

Key words and phrases. Nonlinear hyperbolic equation, periodic in a strip solution.

Remark 1. The conditions (3) and (5) are essential and cannot be weakened. Violation of the condition (3) may result in the loss of global solvability in the whole strip D_{b}. As an example, consider the problem

\frac{∂^{2}u}{∂x∂y} = \frac{∂u}{∂y} − u^{1+ε} − 1, u(x, 0) = 0, u(x + ω, y) = u(x, y),

in the strip D_{b}, where b = 2 ∫_{0}^{+∞} \frac{dξ}{1+ξ^{1+ε}}, ε > 0 is an arbitrary constant. The given problem may have at most one solution (see Theorem 2 below). Therefore if u(x, y) is a solution of the given problem, then u(x, y) = u(y) and it is simultaneously the solution of the Cauchy problem

\frac{du}{dy} = u^{1+ε} + 1, u(0) = 0,

defined on the segment [0, b]. But it is impossible since lim_{y→ \frac{b}{2}} u(y) = +∞.

On the other hand, violation of the condition (5) may result in the loss of solvability of the problem (1),(2). To convince ourselves that is so, consider the problem

\frac{∂^{2}u}{∂x∂y} = y \frac{∂u}{∂y} + 1, u(x, 0) = 0, u(x + ω, y) = u(x, y)

for which all conditions of Theorem 1, except of (5), are fulfilled. Nevertheless, the above problem has no solution. In fact, otherwise we should have

\frac{∂u(x, y)}{∂y} = − \frac{1}{y} for 0 < y ≤ b.

But this contradicts the absolute continuity of u.

Theorem 2. Let the conditions (4) and (5) hold and let there exist nonnegative summable functions c_{0} : [0, ω] × [0, b] → ℝ_{+} and c_{1} : [0, b] → ℝ_{+} such that the inequality

|f(x, y, z_{0}, z_{1}) − f(x, y, \overline{z_{0}}, \overline{z_{1}})| ≤ c_{0}(x, y)|z_{0} − \overline{z_{0}}| + c_{1}(x)|z_{1} − \overline{z_{1}}|

holds on D_{b} × ℝ^{2}. Then the problem (1), (2) has at most one solution.

Finally, consider the case where f(x, y, z_{0}, z_{1}) ≡ f(x, y, z_{0}), i.e., where the equation (1) has the form

\frac{∂^{2}u}{∂x∂y} = f(x, y, u).  (6)

In addition, assume that f has partial derivatives in the second and the third arguments satisfying local Carathéodory conditions. Put

g_{0}(x, y, z) = \frac{∂f(x, y, z)}{∂y} , g_{1}(x, y, z) = \frac{∂f(x, y, z)}{∂z} .

Theorem 3. Let there exist a positive constant l and a measurable function σ : [0, b] → {−1, 1} such that the inequalities

|g_{0}(x, y, z)| ≤ l(1 + |z|), |g_{1}(x, y, z)| ≤ l,

and

∫_{0}^{b} g(s) ds > 0  (7)

hold on [0, ω] × [0, b] × ℝ, where g(x) = ess inf{σ(y)g_{1}(x, t, z) : t ∈ [0, b], z ∈ ℝ}. Then the problem (6), (2) is solvable if and only if

∫_{0}^{ω} f(s, 0, φ(s)) ds = 0.

If, besides, g_{i} (i = 0, 1) are locally Lipschitz continuous in z, then the problem (6), (2) is uniquely solvable.

Remark 2. The condition (7) in Theorem 3 is essential and cannot be neglected. For example, it is obvious that the problem

\frac{∂^{2}u}{∂x∂y} = (1 + u^{2})^{\frac{1}{2}} − 1 + sin( \frac{2π}{ω} x)

u(x, 0) = 0, u(x + ω, y) = u(x, y)

has no solution, although all conditions of Theorem 3, except (7), hold since

g_{1}(x, y, z) = (1 + z^{2})^{− \frac{1}{2}} z.

References

1. L. Cesari, A criterion for the existence in a strip of periodic solutions of hyperbolic partial differential equations. Rend. Circ. Mat. Palermo (2) 14(1965), 95–118.

2. K. Deimling, A Carathéodory theory for systems of integral equations. Ann. Mat. Pura ed Appl. 86(1970), 217–260.

3. I. Kiguradze, Boundary-Value Problems for Systems of Ordinary Differential Equations. Journal of Soviet Mathematics 43(1967), No. 2, 2259–2339.

4. T. Kiguradze, On a boundary value problem for quasilinear hyperbolic system. Russian Acad. Sci. Dokl. Math. 47(1993), 21–25.

5. T. Kiguradze, Some boundary value problems for systems of linear partial differential equations of hyperbolic type. Mem. Differential Equations Math. Phys. 1(1994), 1–144.

6. T. Kiguradze, On bounded and periodic in a strip solutions of quasilinear hyperbolic systems. (Russian) Differentsial’nye Uravnenia 30(1994), No. 10, 1760–1773.

7. T. Kiguradze, On bounded in a strip solutions of quasilinear partial differential equations of hyperbolic type. Applicable Analysis 58(1995), 199–214.

8. V. Lakshmikantham and S. G. Pandit, Periodic solutions of hyperbolic partial differential equations. Comput. Math. 11(1985), No. 1–3, 249–259.

9. Liu Baoping, The integral operator method for finding almost-periodic solutions of nonlinear wave equations. Nonlinear Anal., Theory, Meth. Appl. 11(1987), No. 5, 553–564.

10. S. V. Z̆estkov, On periodic solutions of nonlinear hyperbolic systems in a strip. (Russian) Differentsial’nye Uravneniya 25(1989), No. 10, 1806–1807.

Author’s address:

Faculty of Physics

I. Javakhishvili Tbilisi State University

3, I. Chavchavadze Ave., Tbilisi 380028

Georgia