Mem. Differential Equations Math. Phys. 13(1997), 130–131

J. D. Mirzov

ASYMPTOTIC REPRESENTATIONS OF OSCILLATORY SOLUTIONS OF A NONLINEAR EQUATION

(Reported on October, 6, 1997)

Consider the equation

u′′ + t^{−(n+3)/2}|u|^{n} sign u = 0,  (1)

where t > 0, n > 0. In this note asymptotic formulas are derived for oscillatory solutions of (1) which are somewhat different from those given in [1].

Theorem 1. Let 0 < n < 1. Then for any nontrivial oscillatory soluition u(t) of the equation (1) the equalities

u(t) = t^{1/2}f^{−1}[\sqrt{c} sin((c_{0} + o(1)) ln t)], t → +∞,
u′(t) = (1/2)t^{−1/2}f^{−1}[\sqrt{c} sin((c_{0} + o(1)) ln t)] +
+\sqrt{c}t^{−1/2} cos((c_{0} + o(1)) ln t), t → +∞,

hold, where c_{0} > 0, f^{−1} is the function inverse to

f(z) = sign z\sqrt{2|z|^{1+n}/(1 + n) − z^{2}/4} for |z| < 2^{2/(1−n)}

and

0 < c < ((1 − n)/(1 + n))2^{2(1+n)/(1−n)}.

Proof. By means of the transformation

x(s) = t^{1/2}u′(t), y(s) = t^{−1/2}u(t), s = ln t,

the equation (1) can be written as

x′ = x/2 − |y|^{n} sign y, y′ = x − y/2.  (2)

The nontrivial oscillatory solution of (2) defined by the initial conditions x(0) = \sqrt{c}, y(0) = 0 satisfies

x^{2}(s) − x(s)y(s) + 2|y(s)|^{1+n}/(1 + n) ≡ c.

Moreover,

|x(s)| < 2^{(1+n)/(1−n)}, |y(s)| < 2^{2/(1−n)} for − ∞ < s < +∞.

Introduce the functions

w_{1}(τ) = sign y(s)\sqrt{2|y(s)|^{1+n}/(1 + n) − y^{2}(s)/4},
w_{2}(τ) = x(s) − y(s)/2,

1991 Mathematics Subject Classification. 34C15.

Key words and phrases. Nolinear equation, oscillatory solution, asymptotic representation. where

τ = ∫_{0}^{s} \frac{|y(ϑ)|^{n} − |y(ϑ)|/4}{\sqrt{2|y(ϑ)|^{1+n}/(1 + n) − y^{2}(ϑ)/4}} dϑ.  (3)

If the numbers s_{n} are such that y(s_{n}) = 0, then (2) implies

y′(s_{n}) = x(s_{n}) = ±\sqrt{c} ≠ 0.

Therefore, ε > 0 and δ > 0 can be found such that

|y(s)| ≥ ε_{n}|s − s_{n}| for s ∈ [s_{n} − δ_{n}, s_{n} + δ_{n}],

Hence the integrals ∫_{s_{n}}^{s} |y(ϑ)|^{(n−1)/2}dϑ converge. This means that the improper integral (3) converges for any s.

Taking into account, for instance, [2, p. 235], it can be easily verified that w_{1}(τ), w_{2}(τ) is a solution of the problem

w′_{1} = w_{2}, w′_{2} = −w_{1}, w_{1}(0) = 0, w_{2}(0) = \sqrt{c}.

Therefore,

w_{1}(τ) = \sqrt{c} sin τ, w_{2}(τ) = \sqrt{c} cos τ.

In view of the periodicity of y(s), there exists a finite limit

c_{0} = lim_{s→+∞} \frac{τ(s)}{s} ,

and since τ′(s) > 0, we have c_{0} > 0. ■

Remark 1. The theorem remains true for t → 0^{+}.

Remark 2. Asymptotic representations of oscillatory and nonoscillatory solutions of (1) with n > 1 are given in [3].

References

1. Mirzov J. D., Differentsial’nye Uravneniya 32(1996), No. 11, 1576.

2. Filippov V. V., Solution spaces of ordinary differential equations. (Russian) Moscow University Press, Moscow, 1993.

3. Mirzov J. D., Asymptotic formulas for solutions of an Emden–Fowler equation. (Russian) Trudy FORA, 1997, No. 2, 49–55.

Author’s address:

Adygeya State University

Maykop, Adygeya,

Russia