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Singular internal stabilization of the wave equation. (English) Zbl 0920.35029

The asymptotic behaviour of nonlinear wave equations of the form \[ u''- Au+ g(u')\delta_\gamma= 0\quad\text{on }\Omega\times(0, \infty),\quad u= 0\quad\text{on }\Gamma\times (0,\infty), \]
\[ u(x,0)= u^0(x)\quad\text{on }\Omega,\quad u'(x,0)= u^1(x)\quad\text{on }\Omega \] is considered, where \(\gamma\) is a point in the one-space-dimension case, and \(\gamma\) is a simple closed curve satisfying certain regularity conditions in the two-space-dimension case. It is shown that if \(\gamma\) is a ‘reasonable’ point in the first case and a ‘sufficiently regular’ curve in the second case, then \[ \lim_{t\to\infty} (\| u(t)\|_{H^1(\Omega)}+ \| u'(t)\|_{L^2(\Omega)})= 0. \] It is also shown that the decay to zero is not uniform in energy space. Using the theory of nonharmonic Fourier series, an explicit sharp inequality for the decay rate in the one-space-dimension case is also obtained.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A20 Analyticity in context of PDEs
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