T. Kiguradze

I. Javakhishvili Tbilisi State University, Tbilisi, Georgia

ON INITIAL-BOUNDARY VALUE PROBLEMS IN AN INFINITE STRIP FOR A NONLINEAR HYPERBOLIC EQUATION OF THIRD ORDER

Let I ⊂ ℝ be a compact interval containing zero. For the nonlinear equation

u^{(2,1)} = f(x, t, u, u^{(1,0)}, u^{(2,0)}, u^{(0,1)})  (1)

consider the initial–boundary value problems u(x, 0) = φ(x) for x ∈ [0, +∞),

u(0, t) = ψ(t), sup{|u(x, t)| : x ∈ [0, +∞)} < +∞ for t ∈ I; (2_{1})

u(x, 0) = φ(x) for x ∈ ℝ, sup{|u(x, t)| : x ∈ ℝ} < +∞ for t ∈ I. (2_{2})

Here u^{(j,k)}(x, y) = \frac{∂^{j+k}u(x,y)}{∂x^{j}∂y^{k}}, f : ℝ × I × ℝ^{4} → ℝ is a continuous function, φ : ℝ → ℝ is a twice continuously differentiable function, and ψ : I → ℝ is a continuously differentiable function such that φ(0) = ψ(0).

The linear case of equation (1) arises in study of nonsteady simple shearing flow of second order fluids (c.f. [1]) and also in the theory of seepage of homogeneous fluids through fissured rocks ([2]).

We assume that f satisfies the following three conditions: (i) there exists a continuous function δ : I → ℝ such that (f(x, t, u, v, w, z_{1}) − f(x, t, u, v, w, z_{2})) sgn(z_{1} − z_{2}) ≥ δ(t)|z_{1} − z_{2}|; (ii) f is locally Lipschitz continuous with respect to w; (iii) there exist continuous functions l and g : I → ℝ_{+} such that |f(x, t, u, v, w, z)| ≤ l(t)(|u| + |v| + |w| + |z|) + g(t).

Theorem. Let δ(t) > 0 for t ∈ I. Then problem (1), (2_{k}) (k = 1, 2) is solvable. Moreover an arbitrary solution u_{k} of problem (1), (2_{k}) (k = 1, 2) admits the estimate

∥u_{1}∥_{C^{(2,1)}(ℝ_{+}×I)} ≤ M(∥φ∥_{C^{2}(ℝ_{+})} + ∥ψ∥_{C^{1}(I)} + ∥g∥_{C(I)}) for k = 1,
∥u_{2}∥_{C^{(2,1)}(ℝ×I)} ≤ M(∥φ∥_{C^{2}(ℝ)} + ∥g∥_{C(I)}) for k = 2,

where M > 0 is a constant independent of φ, ψ and g. Furthermore if f is locally Lipschitz continuous with respect to u and v, then problem (1), (2_{k}) (k = 1, 2) is uniquely solvable.

The above results are obtained jointly with Prof. V. Lakshmikantham.

Supported by GRDF (Grant No. 3318).

References

1. B. D. Coleman, R. J. Duffin, and V. J. Mizel, Instability, uniqueness, and nonexistence theorems for the equation u_{t} = u_{xx} − u_{xtx} on a strip. Arch. Rat. Mech. Anal. 19(1965), 100–116.

2. G. Barenblat, I. Zheltov, and I. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(1960), 1286–1303.