{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T13:22:40Z","timestamp":1762262560045,"version":"build-2065373602"},"reference-count":0,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Port. Math."],"published-print":{"date-parts":[[2015,7,23]]},"abstract":"<jats:p>We consider the Cauchy problem for the equation<\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:tex-math>\\text{$u_{t}-\\operatorname{div} \\left( a(x,t) |\\nabla u|^{p(x)-2}\\nabla u\\right) =f(x,t)$ in $S_{T}=\\mathbb{R}^{n}\\times(0,T)$}<\/jats:tex-math>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    with measurable but possibly discontinuous variable exponent\n                    <jats:inline-formula>\n                      <jats:tex-math>p(x):\\,\\mathbb{R}^{n}\\mapsto [p^-,p^+]\\subset (1,\\infty)<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . It is shown that for every\n                    <jats:inline-formula>\n                      <jats:tex-math>u(x,0)\\in L^{2}(\\mathbb{R}^{n})<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>f\\in L^{2}(S_T)<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    the problem has at least one weak solution\n                    <jats:inline-formula>\n                      <jats:tex-math>u\\in C^{0}([0,T];L^{2} _{loc}(\\mathbb{R}^{n}))\\cap L^{2}(S_{T})<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:tex-math>|\\nabla u|^{p(x)}\\in L^{1}(S_{T} )<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We derive sufficient conditions for global boundedness of weak solutions and show that the bounded weak solution is unique.\n                  <\/jats:p>","DOI":"10.4171\/pm\/1961","type":"journal-article","created":{"date-parts":[[2015,7,23]],"date-time":"2015-07-23T17:45:03Z","timestamp":1437673503000},"page":"125-144","source":"Crossref","is-referenced-by-count":2,"title":["On the Cauchy problem for evolution $p(x)$-Laplace equation"],"prefix":"10.4171","volume":"72","author":[{"given":"Stanislav","family":"Antontsev","sequence":"first","affiliation":[{"name":"Universidade de Lisboa, Portugal"}]},{"given":"Sergey","family":"Shmarev","sequence":"additional","affiliation":[{"name":"Universidad de Oviedo, Spain"}]}],"member":"2673","container-title":["Portugaliae Mathematica"],"original-title":[],"link":[{"URL":"http:\/\/www.ems-ph.org\/fulltext\/10.4171\/PM\/1961","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T13:12:35Z","timestamp":1762261955000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.4171\/pm\/1961"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,23]]},"references-count":0,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.4171\/pm\/1961","relation":{},"ISSN":["0032-5155","1662-2758"],"issn-type":[{"type":"print","value":"0032-5155"},{"type":"electronic","value":"1662-2758"}],"subject":[],"published":{"date-parts":[[2015,7,23]]}}}