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Amer. Math. Soc."],"abstract":"<p>\n                    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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha slash x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha \/x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We establish the exponential stability of the semigroup for all positive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , 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.\n                  <\/p>","DOI":"10.1090\/proc\/15063","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T09:50:17Z","timestamp":1582710617000},"page":"4273-4284","source":"Crossref","is-referenced-by-count":3,"special_numbering":"736","title":["The damped wave equation with singular damping"],"prefix":"10.1090","volume":"148","author":[{"given":"Pedro","family":"Freitas","sequence":"first","affiliation":[]},{"given":"Nicolas","family":"Hefti","sequence":"additional","affiliation":[]},{"given":"Petr","family":"Siegl","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2020,5,22]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1137\/070699755","article-title":"The stability of the normal state of superconductors in the presence of electric currents","volume":"40","author":"Almog, Y.","year":"2008","journal-title":"SIAM J. 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