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Some nonlinear wave equations with acoustic boundary conditions. (English) Zbl 0979.35105
The authors consider a boundary value problem for the Carrier equation with nonlinear dissipation and acoustic boundary conditions: $u_{tt}-M(\|u\|^2_{L^2})\Delta u+\beta|u_t|^\alpha u_t=0\text{ in }\Omega\times (0,T),\tag{1}$ where $$\Omega$$ is a bounded region of $$\mathbb{R}^n$$ with smooth boundary $$\Gamma=\Gamma_0\cup\Gamma_1$$. Here $$\Gamma_0,\Gamma_1$$ have nonempty interior. The boundary conditions for (1) are $\begin{cases} u=0\text{ on } \Gamma_0\times (0,T),\\ \rho u_t+f w_{tt}+gw_t+hw=0\text{ on }\Gamma_1\times (0,T),\\ \frac{\partial u}{\partial \eta}=w_t\text{ on }\Gamma_1\times (0,T),\end{cases}\tag{2}$ where $$f(x)$$, $$g(x)$$, and $$h(x)$$ are continuous real functions on $$\overline{\Gamma_1}$$ satisfying $$f(x)> 0$$, $$h(x) > 0$$, and $$g(x) > 0$$ for all $$x\in \overline\Gamma_1$$. In (1) and (2), $$\alpha >1$$, $$\beta$$ and $$\rho$$ are positive constants, and the function $$M\in C^1([0,+\infty);\mathbb{R})$$, satisfies $$0 < m_0\leq M(s)$$ as well as $$|M'(s)\sqrt{s}|\leq m_1M(s)$$ for all $$s\geq 0$$ and some positive constants $$m_0$$ and $$m_1$$. The authors prove global existence of a pair of functions $$\{u(x,t) , w(x,t)\}$$ satisfying (1) and (2) and belonging to the space $$V\cap Y\times L^2(\Gamma)$$ provided $$\alpha> 1$$, and the initial data belong to $$V\cap H^2(\Omega)\times V\cap L^{2\alpha+2}(\Omega),$$ where $$Y=\{v\in H^1(\Omega),\;\Delta v\in L^2(\Omega)\}$$ and $$V=\{u\in H^1(\Omega),\;\delta_0u=0$$ a.e. on $$\Gamma_0\}.$$ Furthermore, if the dimension $$n$$ is either 1 or 2 and $$\alpha > 1$$ or $$n = 3$$ and $$1 < \alpha \leq 2$$ then, they prove uniqueness, regularity results and continuous dependence on the parameters. The results are proved using the Galerkin procedure and the energy method. The article is clearly written.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
Carrier equation; nonlinear dissipation; Galerkin procedure
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##### References:
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