Degenerate Quasilinear Hyperbolic Equation with Strong Damping

§1. Introduction.Let $a(¥cdot)$ be a real valued C¹[0, ¥mathrmo¥mathrmo )-function satisfying (1.1) $a(s)¥geq 0$ for $s¥in[0,$ $¥infty)$ .We consider the initial-boundary value problem (IBVP) for degenerate quasilinear hyperbolic equations (1.2) uₜₜ(x, t)-a(¥int¥Omega|¥nabla u(y, t)|²dy)¥Delta u(x, t)-¥Delta llₜ(x, t)=0 on $¥Omega¥times(0, ¥infty)$ where $¥Omega$ is a bounded domain with its boundary $¥partial¥Omega$ in the n-dimensional Euclidean space Rⁿ .When (1.3) a(s)¥geq a₀>0 for $s¥in[0,$ $¥infty)$ for some constant a₀ instead of (1.1), Eq. (1.2) has a classical solution with null Dirichlet boundary condition and suitable initial data, which is unique and decays exponentially as $ t¥rightarrow¥infty$ .Our aim in this paper is to show the existence, uniqueness, regularity and asymptotic behaviour of the solution to (1.2) under the condition (1.1).We shall remark about the asymptotic behaviour: in general the decay property of the solution to (1.2) cannot be expected for an arbitrary a .In fact, let s₀ be a positive number such that a(s₀)=0 and let u₀(x) satisfy u₀(x)¥in H¥infty(¥Omega) , u₀(x)|¥partial¥Omega=0 and ¥int¥Omega|¥nabla u₀(x)|²dx =s₀ .Then u(x, t)¥equiv u₀(x) is just a solution to (1.2) with initial conditions u(x, 0)=u₀(x) , uₜ(x, 0)=0 and boundary condition u(x, t)|¥partial¥Omega=0 .We note that such u₀(x) 's are infinitely many, and that the above example is appli-