Solutions of the Hyperbolic sine–Gordon Equations

Solutions of Nonlinear Hyperbolic Equations at Resonance

Analytic solutions for the Stephen's inverse problem with local boundary conditions including Elliptic and hyperbolic equations

In this paper, two inverse problems of Stephen kind with local (Dirichlet) boundary conditions are investigated. In the first problem only a part of boundary is unknown and in the second problem, the whole of boundary is unknown. For the both of problems, at first, analytic expressions for unknown boundary are presented, then by using these analytic expressions for unknown boundaries and bounda...

Global solutions to hyperbolic Navier-Stokes equations

We consider a hyperbolicly perturbed Navier-Stokes initial value problem in R, n = 2, 3, arising from using a Cattaneo type relation instead of a Fourier type one in the constitutive equations. The resulting system is a hyperbolic one with quasilinear nonlinearities. The global existence of smooth solutions for small data is proved, and relations to the classical Navier-Stokes systems are discu...

The Quenching of Solutions of Semilinear Hyperbolic Equations

We consider the problem u, U,,x + b (u(x, t)), 0< x < L, >0; u(0, t) u(L, t)=0; u(x, O) ut(x, 0)=0. Assume that b (-oe, A) (0, ee) is continuously differentiable, monotone increasing, convex, and satisfies lim,_.a-b(u)= +ee. We prove that there exist numbers L1 and L2, 0 L2, then a weak solution u (to be defined) quenches in the sense that u reaches A in finite time; if...

The Existence of Carathéodory Solutions of Hyperbolic Functional Differential Equations

We consider the following Darboux problem for the functional differential equation ∂u ∂x∂y (x, y) = f ( x, y, u(x,y), ∂u ∂x (x, y), ∂u ∂y (x, y) ) a.e. in [0, a]× [0, b], u(x, y) = ψ(x, y) on [−a0, a]× [−b0, b]\(0, a]× (0, b], where the function u(x,y) : [−a0, 0] × [−b0, 0] → R k is defined by u(x,y)(s, t) = u(s + x, t + y) for (s, t) ∈ [−a0, 0] × [−b0, 0]. We prove a theorem on existence of th...