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Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term. (English) Zbl 1030.35125

The author studies the global existence as well as the exponential decay (as \(t\) tends to \(\infty \)) of weak solutions to the problem \[ u_{tt}+ \Delta^2u + \lambda u_t = \sum_{i=1}^N\frac{\partial}{\partial x_i}\sigma_i u_{x_i} \quad \text{in }\Omega \times (0, \infty), \]
\[ u=0, \;\;\frac{\partial u}{\partial n} = 0 \quad \text{on }\partial \Omega \times [0,\infty), \]
\[ u(x,0)=u_0(x), \;\;u_t(x,0)=u_1(x), \quad x\in \Omega, \] where \(\lambda \geq 0\) is a constant. Under different assumptions the solutions blow up in finite time.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:

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