{"abstract":[{"text":"Abstract\r\n \r\n An initial-boundary value problem for\r\n \r\n \r\n $$\\begin{aligned} \\left\\{ \\begin{array}{ll}u_{tt} = \\big (\\gamma (\\Theta ) u_{xt}\\big )_x + au_{xx} - \\big (f(\\Theta )\\big )_x, \\qquad & x\\in \\Omega , \\ t>0, \\\\[1mm] \\Theta _t = \\Theta _{xx} + \\gamma (\\Theta ) u_{xt}^2 - f(\\Theta ) u_{xt}, \\qquad & x\\in \\Omega , \\ t>0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n \r\n tt\r\n \r\n \r\n =\r\n \r\n (\r\n \r\n γ\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n u\r\n \r\n xt\r\n \r\n \r\n \r\n \r\n )\r\n \r\n x\r\n \r\n +\r\n a\r\n \r\n u\r\n \r\n xx\r\n \r\n \r\n -\r\n \r\n (\r\n \r\n f\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n \r\n )\r\n \r\n x\r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n x\r\n ∈\r\n Ω\r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n [\r\n 1\r\n m\r\n m\r\n ]\r\n \r\n \r\n Θ\r\n t\r\n \r\n =\r\n \r\n Θ\r\n \r\n xx\r\n \r\n \r\n +\r\n γ\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n u\r\n \r\n xt\r\n \r\n 2\r\n \r\n -\r\n f\r\n \r\n (\r\n Θ\r\n )\r\n \r\n \r\n u\r\n \r\n xt\r\n \r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n x\r\n ∈\r\n Ω\r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered in an open bounded real interval\r\n \r\n \r\n $$\\Omega $$\r\n \r\n Ω\r\n \r\n \r\n \r\n . Under the assumption that\r\n \r\n \r\n $$\\gamma \\in C^0([0,\\infty ))$$\r\n \r\n \r\n γ\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n and\r\n \r\n \r\n $$f\\in C^0([0,\\infty ))$$\r\n \r\n \r\n f\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n are such that\r\n \r\n \r\n $$f(0)=0$$\r\n \r\n \r\n f\r\n (\r\n 0\r\n )\r\n =\r\n 0\r\n \r\n \r\n \r\n \r\n , and\r\n \r\n \r\n $$k_\\gamma \\le \\gamma \\le K_\\gamma $$\r\n \r\n \r\n \r\n k\r\n γ\r\n \r\n ≤\r\n γ\r\n ≤\r\n \r\n K\r\n γ\r\n \r\n \r\n \r\n \r\n \r\n as well as\r\n \r\n \r\n $$\\begin{aligned} |f(\\xi )| \\le K_f \\cdot (\\xi +1)^\\alpha \\qquad \\hbox {for all } \\xi \\ge 0 \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n |\r\n f\r\n \r\n (\r\n ξ\r\n )\r\n \r\n |\r\n \r\n ≤\r\n \r\n K\r\n f\r\n \r\n ·\r\n \r\n \r\n (\r\n ξ\r\n +\r\n 1\r\n )\r\n \r\n α\r\n \r\n \r\n for all\r\n \r\n ξ\r\n ≥\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n with some\r\n \r\n \r\n $$k_\\gamma>0, K_\\gamma>0, K_f>0$$\r\n \r\n \r\n \r\n k\r\n γ\r\n \r\n >\r\n 0\r\n ,\r\n \r\n K\r\n γ\r\n \r\n >\r\n 0\r\n ,\r\n \r\n K\r\n f\r\n \r\n >\r\n 0\r\n \r\n \r\n \r\n \r\n and\r\n \r\n \r\n $$\\alpha <\\frac{3}{2}$$\r\n \r\n \r\n α\r\n <\r\n \r\n 3\r\n 2\r\n \r\n \r\n \r\n \r\n \r\n , for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. By particularly covering the thermodynamically consistent choice\r\n \r\n \r\n $$f\\equiv id$$\r\n \r\n \r\n f\r\n ≡\r\n i\r\n d\r\n \r\n \r\n \r\n \r\n of predominant physical relevance, this appears to go beyond previous related literature which seems to either rely on independence of\r\n \r\n \r\n $$\\gamma $$\r\n \r\n γ\r\n \r\n \r\n \r\n on\r\n \r\n \r\n $$\\Theta $$\r\n \r\n Θ\r\n \r\n \r\n \r\n , or to operate on finite time intervals.\r\n ","lang":"eng"}],"language":[{"iso":"eng"}],"doi":"10.1007/s00033-025-02582-y","publication_status":"published","author":[{"last_name":"Winkler","full_name":"Winkler, Michael","id":"31496","first_name":"Michael"}],"status":"public","type":"journal_article","volume":76,"publication_identifier":{"issn":["0044-2275","1420-9039"]},"intvolume":" 76","date_created":"2025-12-18T19:03:19Z","date_updated":"2025-12-18T20:13:25Z","year":"2025","title":"Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities","citation":{"ieee":"M. Winkler, “Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities,” Zeitschrift für angewandte Mathematik und Physik, vol. 76, no. 5, Art. no. 192, 2025, doi: 10.1007/s00033-025-02582-y.","mla":"Winkler, Michael. “Large-Data Solutions in One-Dimensional Thermoviscoelasticity Involving Temperature-Dependent Viscosities.” Zeitschrift Für Angewandte Mathematik Und Physik, vol. 76, no. 5, 192, Springer Science and Business Media LLC, 2025, doi:10.1007/s00033-025-02582-y.","bibtex":"@article{Winkler_2025, title={Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities}, volume={76}, DOI={10.1007/s00033-025-02582-y}, number={5192}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }","ama":"Winkler M. Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities. Zeitschrift für angewandte Mathematik und Physik. 2025;76(5). doi:10.1007/s00033-025-02582-y","short":"M. Winkler, Zeitschrift Für Angewandte Mathematik Und Physik 76 (2025).","chicago":"Winkler, Michael. “Large-Data Solutions in One-Dimensional Thermoviscoelasticity Involving Temperature-Dependent Viscosities.” Zeitschrift Für Angewandte Mathematik Und Physik 76, no. 5 (2025). https://doi.org/10.1007/s00033-025-02582-y.","apa":"Winkler, M. (2025). Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities. Zeitschrift Für Angewandte Mathematik Und Physik, 76(5), Article 192. https://doi.org/10.1007/s00033-025-02582-y"},"_id":"63250","user_id":"31496","publication":"Zeitschrift für angewandte Mathematik und Physik","publisher":"Springer Science and Business Media LLC","article_number":"192","issue":"5"}