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The global existence of small amplitude solutions to the nonlinear acoustic wave equation. (English) Zbl 0794.35108
The nonlinear wave equation in a viscous conducting fluid, modeled by \[ \partial_{tt}\varphi-c^ 2_ 0\Delta\varphi=\partial_ t\{|\nabla\varphi|^ 2+b\Delta\varphi+a|\partial_ t\varphi|^ 2\}\quad\text{ in }\Omega\times[0,\infty) \] is considered, where \(a,b,c^ 2_ 0>0\), and \(\Omega=\mathbb{R}^ n\), or \(\Omega\) is a bounded domain with smooth boundary assuming Dirichlet boundary conditions for \(\varphi\). The existence of a unique global solution to the corresponding initial (-boundary) value problem is shown for small data \(\varphi(t=0)\), \(\varphi_ t(t=0)\), if \(n=1,2\) or 3. If \(\Omega\) is bounded, the exponential decay is also proved. The main tool are energy estimates.
Reviewer: R.Racke (Bonn)

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35Q35 PDEs in connection with fluid mechanics
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