Existence, uniqueness and asymptotic
behavior of solutions of singular second
order ODE
Jakub Stryja
Katedra matematiky a deskriptivnı́ geometrie, VŠB-TU Ostrava
17. listopadu 15/2172, 708 33 Ostrava-Poruba
E-mail: [email protected]
Abstrakt: Přı́spěvek se zabývá existencı́, jeznoznačnostı́ a asymptotickým chovánı́m řešenı́ rovnice
0
(p(t)u0 (t)) = p(t)f (u(t))
na polopřı́mce h 0; ∞ ). Funkce f lokálně na R splňuje Lipchitzovu podmı́nku a má
alespoň dva nulové body, p(0) = 0, dále je funkce p spojitá na h 0; ∞ ) a má kladnou
spojitou derivaci na (0; ∞).
Abstract: We investigate the singular differential equation
0
(p(t)u0 (t)) = p(t)f (u(t))
on the half-line h 0; ∞). Function f satisfies the local Lipschitz condition on R and
has at least two simple zeros. The function p is continuous on h 0; ∞ ) and has
a positive continuous derivative on (0; ∞) and p(0) = 0.
1
Introduction
We investigate a singular boundary value problem motivated by some models used
in nonlinear field theory or in the Cahn-Hilliard theory in hydrodynamics. If ρ is
the density, µ(ρ) the chemical potential of a non-homogeneous fluid and the motion
of the fluid is zero, then the state of the fluid in RN is described by the equation
γ∆ρ = µ(ρ) − µ0 ,
(1)
where γ and µ0 are suitable constants. When we search for a solution with the
spherical symmetry, then equation (1) is reduced to the ordinary differential equation
N −1 0
00
ρ = µ(ρ) − µ0 , r ∈ (0; ∞) .
(2)
γ ρ +
r
Equation (2) with the boundary conditions
ρ0 (0) = 0 ,
lim ρ(r) = ρ` > 0
r→∞
(3)
describe the formation of microscopic bubbles in a fluid, in particular, vapor inside
liquid. The first condition in (3) follows from central symmetry and it is necessary
for the smoothness of solutions of the singular equation (2) at r = 0. The second
condition in (3) means the bubble is surrounded by an external liquid with density
ρ` .
Let N = 3. In the simplest model of non-homogeneous fluid, problem (2), (3) is
reduced to the form
0
t2 u0 = 4λ2 t2 (u + 1)u(u − ξ) ,
(4)
u0 (0) = 0 ,
u(∞) = ξ ,
(5)
where λ ∈ (0; ∞) and ξ ∈ (0; 1) are parameters. Many important physical properties
of the bubbles depend on the existence of an increasing solution of the problem (4),
(5) with just one zero. In particular, the gas density inside the bubble, the bubble
radius and the surface tension.
2
Formulation of the problem
We investigate generalizations of the problem (4), (5). We study the equation
0
(p(t)u0 (t)) = p(t)f (u(t))
(6)
on the half-line h 0; ∞ ), where
f ∈ Liploc (R) ,
p(0) = 0 ,
p ∈ C h 0; ∞) ∩ C 1 (0; ∞) ,
p0 (t) > 0 for t > 0 ,
p0 (t)
=0.
t→∞ p(t)
lim
(7)
(8)
There exist L0 < 0 (L0 = −∞ is possible), L > 0, CL > 0 such that
xf (x) < 0 for x ∈ (L0 ; 0) ∪ (0; L) ,
(9)
0 ≤ f (x) ≤ CL for x ≥ L .
(10)
Figure 1: Example of function f .
Example 2. An example of the function p satisfying (7), (8) is p(t) = t2 , t ∈ h0; ∞).
Remark 3. Equation (6) is singular at t = 0 because p(0) = 0 and
0
(p(t)u0 (t)) = p(t)f (u(t)) ,
p(t)u00 (t) + p0 (t)u0 (t) = p(t)f (u(t)) ,
u00 (t) +
Z
0
ε
p0 (t) 0
u (t) − f (u(t)) = 0 ,
p(t)
p0 (t)
dt = ln(p(ε)) − lim ln(p(t)) = ∞ .
t→0+
p(t)
Definition 4. A function u ∈ C 1 h 0; ∞ ) which has a continuous second derivative
on (0; ∞) and satisfies equation (6) for all t ∈ (0; ∞) is called solution of (6).
Consider B < 0 and the initial conditions
u(0) = B ,
u0 (0) = 0 .
(11)
Remark 5. Consider a solution u of equation (6). Since u ∈ C 1 h 0; ∞ ), we have
u(0), u0 (0) ∈ R, and the assumption p(0) = 0 yields p(0)u0 (0) = 0. We can find
M > 0 and δ > 0 such that |f (u(t))| ≤ M for t ∈ (0; δ). Integrating equation (6)
and using the fact, that p is increasing, we get
Z t
Z t
M
1
0
p(s)f (u(s)) ds ≤
p(s) ds ≤ M t for t ∈ (0; δ) .
|u (t)| = p(t) 0
p(t) 0
Consequently, the condition u0 (0) = 0 is necessary for each solution u of equation
(6). Therefore the set of all solutions of equation (6) forms a one-parameter system
of functions u satisfying u(0) = A, A ∈ R.
3
Existence and uniqueness
Theorem 6 (Existence and uniqueness). Assume that (7), (8), (9), and (10)
hold and let B ∈ (L0 ; 0). Then problem (6), (11) has a unique solution u, and
moreover the solution u satisfies
u(t) ≥ B for t ∈ h0; ∞ ) .
4
Asymptotic behavior
Consider such a solution u and denote
usup = sup{u(t) : t ∈ h0; ∞ )} .
Definition 7. If usup < L (usup = L or usup > L), then u is called a damped solution
(a homoclinic solution or an escape solution) of the problem (6), (11).
Figure 2: Three types of solutions.
Definition 8. A damped solution is oscillatory, if it has an unbounded set of
isolated zeros.
Let us put
Z
x
f (z) dz for x ∈ R .
F (x) = −
0
Due to (7), (9) the function F is continuous on R, decreasing and positive on
(L0 ; 0), increasing and positive on (0; L). Therefore we can define B < 0 by
B = inf{B0 ∈ (L0 ; 0) : F (B) < F (L) ∀B ∈ (B0 ; 0)}
(12)
(B = −∞ is possible).
Theorem 9 (Existence of damped solutions). Assume that (7), (8) (9), and
(10) hold. Let B be given
by (12), and assume that u is a solution of the problem
(6), (11) with B ∈ B; 0 . Then u is a damped solution.
Now we bring additional conditions for f and p under which the equation has
oscillatory solutions with decreasing amplitudes.
f (x)
<0,
x→0+ x
00 p (t) lim sup 0 < ∞ .
p (t)
t→∞
f (x)
<0,
x→0− x
lim
p ∈ C 2 (0; ∞) ,
lim
(13)
(14)
Theorem 10. Assume that (7), (8), (9), (10), (13), and (14) hold. Let u be
a solution of the problem (6), (11) with B ∈ (L0 ; 0). If u is a damped solution, then
u is oscillatory and its amplitudes are decreasing.
The next assertion follows from Theorems 9 and 10.
Theorem 11 (Existence of oscillatory solutions). Assume that (7), (8), (9),
(10), (13), and (14) hold. Let B
be given by (12) and let u be a solution of the
problem (6), (11) with B ∈ B; 0 . Then u is an oscillatory solution with decreasing
amplitudes.
Remark 12. The assumption (10) in Theorem 11 can be omitted, because it has no
influence on the existence of oscillatory solutions. It follows from the fact that (10)
imposes conditions on the function values of the function f for arguments greater
than L; however, the function values of oscillatory solutions are lower than this
constant L. This condition (used only in Theorem 6) guaranteed the existence of
solution of each problem (6), (11) for each B < 0 on the whole half-line, which
simplified the investigation of the problem.
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