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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)

##### MSC:
 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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