T. Jangveladze and Z. Kiguradze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON A SYSTEM OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

In the cylinder Q = (0, 1)×(0, ∞), we consider the initial boundary value problem

\frac{∂U}{∂t} = \frac{∂}{∂x} [a(S) \frac{∂U}{∂x} ], \frac{∂V}{∂t} = \frac{∂}{∂x} [a(S) \frac{∂V}{∂x} ], (x, t) ∈ Q,  (1)
U(0, t) = U(1, t) = V (0, t) = V (1, t) = 0, t ≥ 0,  (2)
U(x, 0) = U_{0}(x), V (x, 0) = V_{0}(x), x ∈ [0, 1],  (3)

where

S(t) = ∫_{0}^{t} ∫_{0}^{1} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}] dx dτ,  (4)

or

S(x, t) = ∫_{0}^{t} [(\frac{∂U}{∂x} )^{2} + (\frac{∂V}{∂x} )^{2}] dτ.  (5)

Theorem 1. If a(s) ≥ a_{0} = const > 0, U_{0}, V_{0} ∈ W^{∘} _{2}^{1}(0, 1), then for problem (1)–(4) the following estimate is true

∥U∥_{W_{2}^{1}} + ∥V ∥_{W_{2}^{1}} ≤ Ce^{− \frac{a_{0}}{2} t}.

Consider the boundary conditions

U(0, t) = V (0, t) = 0, U(1, t) = ψ_{1}, V (1, t) = ψ_{2}, t ≥ 0.  (6)

Then the following statement is true.

Theorem 2. If a(S) = (1 + S)^{p}, p ∈ (−1/2, 0) or p ∈ (0, 1], U_{0}(0) = V_{0}(0) = 0, U_{0}(1) = ψ_{1}, V_{0}(1) = ψ_{2}, ψ_{1}^{2} + ψ_{2}^{2} ≠ 0, U_{0}, V_{0} ∈ W_{2}^{2}(0, 1), then for problem (1)–(4) and (1), (3), (5), (6) the following estimates are true:

\frac{∂U(x, t)}{∂x} = ψ_{1} + O(t^{−1−p}), \frac{∂V (x, t)}{∂x} = ψ_{2} + O(t^{−1−p}),

\frac{∂U(x, t)}{∂t} = O(t^{−1}), \frac{∂V (x, t)}{∂t} = O(t^{−1})

as t → ∞, uniformly in x on [0, 1].

The numerical solutions of problems (1)–(4) and (1), (3), (5), (6) are also obtained. The numerical experiments are agreed with the theoretical conclusions.