The damped wave equation with singular damping
Copyright policy )The damped wave equation with singular damping
We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form α / x \alpha /x , α > 0 \alpha >0 . We establish the exponential stability of the semigroup for all positive α \alpha , and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.
- University of Bern Switzerland
- University of Lisbon Portugal
- Queen's University Belfast United Kingdom
- Instituto Superior de Espinho Portugal
Singular hyperbolic equations, Asymptotic behavior of solutions to PDEs, finite-time extinction, FOS: Physical sciences, Estimates of eigenvalues in context of PDEs, Mathematical Physics (math-ph), 510, Functional Analysis (math.FA), Mathematics - Spectral Theory, Mathematics - Functional Analysis, Mathematics - Analysis of PDEs, empty spectrum, FOS: Mathematics, Laguerre polynomials, Initial-boundary value problems for second-order hyperbolic equations, Spectral Theory (math.SP), 35P15, 35L05, Mathematical Physics, Analysis of PDEs (math.AP)
Singular hyperbolic equations, Asymptotic behavior of solutions to PDEs, finite-time extinction, FOS: Physical sciences, Estimates of eigenvalues in context of PDEs, Mathematical Physics (math-ph), 510, Functional Analysis (math.FA), Mathematics - Spectral Theory, Mathematics - Functional Analysis, Mathematics - Analysis of PDEs, empty spectrum, FOS: Mathematics, Laguerre polynomials, Initial-boundary value problems for second-order hyperbolic equations, Spectral Theory (math.SP), 35P15, 35L05, Mathematical Physics, Analysis of PDEs (math.AP)
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