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On some nonlinear wave equations. II: Global existence and energy decay of solutions. (English) Zbl 0783.35038

The authors consider the mixed problem with restoring and damping terms for the hyperbolic type equation \[ u_{tt}-M \left( \int_ \Omega | \nabla u |^ 2 dx \right) \Delta u+\delta | u |^ \alpha u+\gamma u_ t=f \quad \text{in } \Omega \times [0,\infty) \tag{1} \] and data (2) \(u=0\) on \(\Gamma \times[0,\infty)\), (3) \(u(x,0)=u^ 0(x)\), \(u_ t(x,0)=u^ 1(x)\) in \(\Omega\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) with smooth boundary \(\Gamma\); \(\delta>0\), \(\alpha \geq 0\) and \(\gamma>0\) are given constants and \(M \in C^ 1\bigl( [0,\infty) \bigr)\) is a positive function.
The present problem is an extension of part I (reviewed above) by the damping term \(\gamma u_ t\). Again by Galerkin’s method they prove the global existence and the decay rates of solutions to zero as \(t \to \infty\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0783.35037
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