# ON EXISTENCE OF SOLUTIONS OF DEGENERATE WAVE EQUATIONS WITH NONLINEAR DAMPING TERMS

• Park, Jong-Yeoul (Department of Mathematics Pusan National University) ;
• Bae, Jeong-Ja (Department of Mathematics Pusan National University)
• Published : 1998.05.01
• 64 7

#### Abstract

In this paper, we consider the existence and asymptotic behavior of solutions of the following problem: $u_{tt}$ -(t, x) - (∥∇u(t, x)∥(equation omitted) ＋ ∥∇v(t, x) (equation omitted)$^{\gamma}$ $\Delta$u(t, x)＋$\delta$$u_{t} (t, x)│sup p-1/ u_{t} (t, x) = \mu│u(t, x) ^{q-1}u(t, x), x\in$$\Omega$, t$\in$[0, T], $v_{tt}$ (t, x) - (∥∇uu(t, x) (equation omitted) ＋ ∥∇v(t, x) (equation omitted)sup ${\gamma}$/ $\Delta$v(t, x)＋$\delta$$v_{t} (t, x)│sup p-1/ u_{t} (t, x) = \mu u(t, x) ^{q-1}u(t, x), x\in$$\Omega$, t$\in$[0, T], u(0, x) = $u_{0}$ (x), $u_{t}$ (0, x) = $u_1$(x), x$\in$$\Omega, u(0, x) = v_{0} (x), v_{t} (0, x) = v_1(x), x\in$$\Omega$, u│∂$\Omega$=v│∂$\Omega$=0 T > 0, q > 1, p $\geq$1, $\delta$ > 0, $\mu$ $\in$ R, ${\gamma}$ $\geq$ 1 and $\Delta$ is the Laplacian in $R^{N}$.X> N/.

#### Keywords

existence and uniqueness;asymptotic behavior;degenerate wave equation;Galerkin method