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A convergent algorithm for forced mean curvature flow driven by diffusion on the surface. <i>Interfaces and Free Boundaries</i>, <i>22</i>(4), 443–464. <a href=\"https://doi.org/10.4171/ifb/446\">https://doi.org/10.4171/ifb/446</a>"},"publication_identifier":{"issn":["1463-9963"]},"publication_status":"published","doi":"10.4171/ifb/446","date_updated":"2024-04-03T09:21:02Z","volume":22,"author":[{"first_name":"Balázs","last_name":"Kovács","orcid":"0000-0001-9872-3474","id":"100441","full_name":"Kovács, Balázs"},{"full_name":"Li, Buyang","last_name":"Li","first_name":"Buyang"},{"last_name":"Lubich","full_name":"Lubich, Christian","first_name":"Christian"}],"status":"public","type":"journal_article","_id":"45952","department":[{"_id":"841"}],"user_id":"100441","year":"2020","issue":"4","title":"A convergent algorithm for forced mean curvature flow driven by diffusion on the surface","publisher":"European Mathematical Society - EMS - Publishing House GmbH","date_created":"2023-07-10T11:42:14Z","publication":"Interfaces and Free Boundaries","keyword":["Applied Mathematics"],"language":[{"iso":"eng"}]},{"language":[{"iso":"eng"}],"department":[{"_id":"10"}],"user_id":"49063","_id":"34632","status":"public","editor":[{"first_name":"Fred J.","full_name":"Hickernell, Fred J.","last_name":"Hickernell"},{"first_name":"Peter","last_name":"Kritzer","full_name":"Kritzer, Peter"}],"publication":"Multivariate Algorithms and Information-Based Complexity","type":"book_chapter","title":"RBF-based penalized least-squares approximation of noisy scattered data on the sphere","author":[{"last_name":"Hesse","orcid":"0000-0003-4125-1941","full_name":"Hesse, Kerstin","id":"42608","first_name":"Kerstin"}],"date_created":"2022-12-20T17:34:38Z","date_updated":"2024-04-03T11:03:30Z","publisher":"De Gruyter","page":"33-42 ","citation":{"chicago":"Hesse, Kerstin. “RBF-Based Penalized Least-Squares Approximation of Noisy Scattered Data on the Sphere.” In <i>Multivariate Algorithms and Information-Based Complexity</i>, edited by Fred J. Hickernell and Peter Kritzer, 33–42. Berlin/Boston: De Gruyter, 2020.","ieee":"K. Hesse, “RBF-based penalized least-squares approximation of noisy scattered data on the sphere,” in <i>Multivariate Algorithms and Information-Based Complexity</i>, F. J. Hickernell and P. Kritzer, Eds. Berlin/Boston: De Gruyter, 2020, pp. 33–42.","ama":"Hesse K. RBF-based penalized least-squares approximation of noisy scattered data on the sphere. In: Hickernell FJ, Kritzer P, eds. <i>Multivariate Algorithms and Information-Based Complexity</i>. De Gruyter; 2020:33-42.","bibtex":"@inbook{Hesse_2020, place={Berlin/Boston}, title={RBF-based penalized least-squares approximation of noisy scattered data on the sphere}, booktitle={Multivariate Algorithms and Information-Based Complexity}, publisher={De Gruyter}, author={Hesse, Kerstin}, editor={Hickernell, Fred J. and Kritzer, Peter}, year={2020}, pages={33–42} }","short":"K. Hesse, in: F.J. Hickernell, P. Kritzer (Eds.), Multivariate Algorithms and Information-Based Complexity, De Gruyter, Berlin/Boston, 2020, pp. 33–42.","mla":"Hesse, Kerstin. “RBF-Based Penalized Least-Squares Approximation of Noisy Scattered Data on the Sphere.” <i>Multivariate Algorithms and Information-Based Complexity</i>, edited by Fred J. Hickernell and Peter Kritzer, De Gruyter, 2020, pp. 33–42.","apa":"Hesse, K. (2020). RBF-based penalized least-squares approximation of noisy scattered data on the sphere. In F. J. Hickernell &#38; P. Kritzer (Eds.), <i>Multivariate Algorithms and Information-Based Complexity</i> (pp. 33–42). De Gruyter."},"place":"Berlin/Boston","year":"2020","publication_identifier":{"isbn":["9783110633115"]}},{"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","date_updated":"2024-04-05T13:22:19Z","date_created":"2024-04-05T09:05:11Z","author":[{"first_name":"Mohammad","last_name":"Soleymani","full_name":"Soleymani, Mohammad"},{"full_name":"Santamaria, Ignacio","last_name":"Santamaria","first_name":"Ignacio"},{"first_name":"Peter J.","full_name":"Schreier, Peter J.","last_name":"Schreier"}],"volume":69,"title":"Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments","doi":"10.1109/tvt.2020.3015558","publication_status":"published","publication_identifier":{"issn":["0018-9545","1939-9359"]},"issue":"10","year":"2020","citation":{"apa":"Soleymani, M., Santamaria, I., &#38; Schreier, P. J. (2020). Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments. <i>IEEE Transactions on Vehicular Technology</i>, <i>69</i>(10), 11632–11645. <a href=\"https://doi.org/10.1109/tvt.2020.3015558\">https://doi.org/10.1109/tvt.2020.3015558</a>","mla":"Soleymani, Mohammad, et al. “Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments.” <i>IEEE Transactions on Vehicular Technology</i>, vol. 69, no. 10, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 11632–45, doi:<a href=\"https://doi.org/10.1109/tvt.2020.3015558\">10.1109/tvt.2020.3015558</a>.","short":"M. Soleymani, I. Santamaria, P.J. Schreier, IEEE Transactions on Vehicular Technology 69 (2020) 11632–11645.","bibtex":"@article{Soleymani_Santamaria_Schreier_2020, title={Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments}, volume={69}, DOI={<a href=\"https://doi.org/10.1109/tvt.2020.3015558\">10.1109/tvt.2020.3015558</a>}, number={10}, journal={IEEE Transactions on Vehicular Technology}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Soleymani, Mohammad and Santamaria, Ignacio and Schreier, Peter J.}, year={2020}, pages={11632–11645} }","chicago":"Soleymani, Mohammad, Ignacio Santamaria, and Peter J. Schreier. “Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments.” <i>IEEE Transactions on Vehicular Technology</i> 69, no. 10 (2020): 11632–45. <a href=\"https://doi.org/10.1109/tvt.2020.3015558\">https://doi.org/10.1109/tvt.2020.3015558</a>.","ieee":"M. Soleymani, I. Santamaria, and P. J. Schreier, “Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments,” <i>IEEE Transactions on Vehicular Technology</i>, vol. 69, no. 10, pp. 11632–11645, 2020, doi: <a href=\"https://doi.org/10.1109/tvt.2020.3015558\">10.1109/tvt.2020.3015558</a>.","ama":"Soleymani M, Santamaria I, Schreier PJ. Improper Gaussian Signaling for the $K$-User MIMO Interference Channels With Hardware Impairments. <i>IEEE Transactions on Vehicular Technology</i>. 2020;69(10):11632-11645. doi:<a href=\"https://doi.org/10.1109/tvt.2020.3015558\">10.1109/tvt.2020.3015558</a>"},"page":"11632-11645","intvolume":"        69","_id":"53270","user_id":"67076","department":[{"_id":"263"}],"keyword":["Electrical and Electronic Engineering","Computer Networks and Communications","Aerospace Engineering","Automotive Engineering"],"language":[{"iso":"eng"}],"type":"journal_article","publication":"IEEE Transactions on Vehicular Technology","status":"public"},{"year":"2020","citation":{"ama":"Soleymani M, Santamaria I, Maham B, Schreier PJ. Rate Region of the K-user MIMO Interference Channel with Imperfect Transmitters. In: <i>2020 28th European Signal Processing Conference (EUSIPCO)</i>. IEEE; 2020. doi:<a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">10.23919/eusipco47968.2020.9287450</a>","chicago":"Soleymani, Mohammad, Ignacio Santamaria, Behrouz Maham, and Peter J. Schreier. “Rate Region of the K-User MIMO Interference Channel with Imperfect Transmitters.” In <i>2020 28th European Signal Processing Conference (EUSIPCO)</i>. IEEE, 2020. <a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">https://doi.org/10.23919/eusipco47968.2020.9287450</a>.","ieee":"M. Soleymani, I. Santamaria, B. Maham, and P. J. Schreier, “Rate Region of the K-user MIMO Interference Channel with Imperfect Transmitters,” 2020, doi: <a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">10.23919/eusipco47968.2020.9287450</a>.","bibtex":"@inproceedings{Soleymani_Santamaria_Maham_Schreier_2020, title={Rate Region of the K-user MIMO Interference Channel with Imperfect Transmitters}, DOI={<a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">10.23919/eusipco47968.2020.9287450</a>}, booktitle={2020 28th European Signal Processing Conference (EUSIPCO)}, publisher={IEEE}, author={Soleymani, Mohammad and Santamaria, Ignacio and Maham, Behrouz and Schreier, Peter J.}, year={2020} }","mla":"Soleymani, Mohammad, et al. “Rate Region of the K-User MIMO Interference Channel with Imperfect Transmitters.” <i>2020 28th European Signal Processing Conference (EUSIPCO)</i>, IEEE, 2020, doi:<a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">10.23919/eusipco47968.2020.9287450</a>.","short":"M. Soleymani, I. Santamaria, B. Maham, P.J. Schreier, in: 2020 28th European Signal Processing Conference (EUSIPCO), IEEE, 2020.","apa":"Soleymani, M., Santamaria, I., Maham, B., &#38; Schreier, P. J. (2020). Rate Region of the K-user MIMO Interference Channel with Imperfect Transmitters. <i>2020 28th European Signal Processing Conference (EUSIPCO)</i>. <a href=\"https://doi.org/10.23919/eusipco47968.2020.9287450\">https://doi.org/10.23919/eusipco47968.2020.9287450</a>"},"publication_status":"published","title":"Rate Region of the K-user MIMO Interference Channel with Imperfect Transmitters","doi":"10.23919/eusipco47968.2020.9287450","date_updated":"2024-04-05T13:21:59Z","publisher":"IEEE","author":[{"first_name":"Mohammad","last_name":"Soleymani","full_name":"Soleymani, Mohammad"},{"last_name":"Santamaria","full_name":"Santamaria, Ignacio","first_name":"Ignacio"},{"last_name":"Maham","full_name":"Maham, Behrouz","first_name":"Behrouz"},{"first_name":"Peter J.","full_name":"Schreier, Peter J.","last_name":"Schreier"}],"date_created":"2024-04-05T09:05:01Z","status":"public","type":"conference","publication":"2020 28th European Signal Processing Conference (EUSIPCO)","language":[{"iso":"eng"}],"_id":"53269","user_id":"67076","department":[{"_id":"263"}]},{"language":[{"iso":"eng"}],"user_id":"56070","_id":"53305","status":"public","publication":"2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)","type":"conference","doi":"10.1109/etfa46521.2020.9211880","title":"DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms","author":[{"full_name":"Mohammadi, Hassan Ghasemzadeh","last_name":"Mohammadi","first_name":"Hassan Ghasemzadeh"},{"full_name":"Arshad, Rahil","last_name":"Arshad","first_name":"Rahil"},{"full_name":"Rautmare, Sneha","last_name":"Rautmare","first_name":"Sneha"},{"first_name":"Suraj","full_name":"Manjunatha, Suraj","last_name":"Manjunatha"},{"first_name":"Maurice","last_name":"Kuschel","full_name":"Kuschel, Maurice","id":"56070"},{"full_name":"Jentzsch, Felix Paul","last_name":"Jentzsch","first_name":"Felix Paul"},{"first_name":"Marco","full_name":"Platzner, Marco","last_name":"Platzner"},{"first_name":"Alexander","last_name":"Boschmann","full_name":"Boschmann, Alexander"},{"full_name":"Schollbach, Dirk","last_name":"Schollbach","first_name":"Dirk"}],"date_created":"2024-04-05T14:41:44Z","date_updated":"2024-04-05T14:50:35Z","publisher":"IEEE","citation":{"ieee":"H. G. Mohammadi <i>et al.</i>, “DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms,” 2020, doi: <a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">10.1109/etfa46521.2020.9211880</a>.","chicago":"Mohammadi, Hassan Ghasemzadeh, Rahil Arshad, Sneha Rautmare, Suraj Manjunatha, Maurice Kuschel, Felix Paul Jentzsch, Marco Platzner, Alexander Boschmann, and Dirk Schollbach. “DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms.” In <i>2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)</i>. IEEE, 2020. <a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">https://doi.org/10.1109/etfa46521.2020.9211880</a>.","ama":"Mohammadi HG, Arshad R, Rautmare S, et al. DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms. In: <i>2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)</i>. IEEE; 2020. doi:<a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">10.1109/etfa46521.2020.9211880</a>","apa":"Mohammadi, H. G., Arshad, R., Rautmare, S., Manjunatha, S., Kuschel, M., Jentzsch, F. P., Platzner, M., Boschmann, A., &#38; Schollbach, D. (2020). DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms. <i>2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)</i>. <a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">https://doi.org/10.1109/etfa46521.2020.9211880</a>","mla":"Mohammadi, Hassan Ghasemzadeh, et al. “DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms.” <i>2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)</i>, IEEE, 2020, doi:<a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">10.1109/etfa46521.2020.9211880</a>.","bibtex":"@inproceedings{Mohammadi_Arshad_Rautmare_Manjunatha_Kuschel_Jentzsch_Platzner_Boschmann_Schollbach_2020, title={DeepWind: An Accurate Wind Turbine Condition Monitoring Framework via Deep Learning on Embedded Platforms}, DOI={<a href=\"https://doi.org/10.1109/etfa46521.2020.9211880\">10.1109/etfa46521.2020.9211880</a>}, booktitle={2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA)}, publisher={IEEE}, author={Mohammadi, Hassan Ghasemzadeh and Arshad, Rahil and Rautmare, Sneha and Manjunatha, Suraj and Kuschel, Maurice and Jentzsch, Felix Paul and Platzner, Marco and Boschmann, Alexander and Schollbach, Dirk}, year={2020} }","short":"H.G. Mohammadi, R. Arshad, S. Rautmare, S. Manjunatha, M. Kuschel, F.P. Jentzsch, M. Platzner, A. Boschmann, D. Schollbach, in: 2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA), IEEE, 2020."},"year":"2020","publication_status":"published"},{"type":"journal_article","publication":"Journal of Dynamics and Differential Equations","status":"public","abstract":[{"text":"<jats:title>Abstract</jats:title><jats:p>The Cauchy problem in <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {R}}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mi>R</mml:mi>\r\n                    </mml:mrow>\r\n                    <mml:mi>n</mml:mi>\r\n                  </mml:msup>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n                    <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, for the parabolic equation <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:mrow>\r\n                            <mml:msub>\r\n                              <mml:mi>u</mml:mi>\r\n                              <mml:mi>t</mml:mi>\r\n                            </mml:msub>\r\n                            <mml:mo>=</mml:mo>\r\n                            <mml:msup>\r\n                              <mml:mi>u</mml:mi>\r\n                              <mml:mi>p</mml:mi>\r\n                            </mml:msup>\r\n                            <mml:mi>Δ</mml:mi>\r\n                            <mml:mi>u</mml:mi>\r\n                            <mml:mspace />\r\n                            <mml:mspace />\r\n                            <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n                              <mml:mo>⋆</mml:mo>\r\n                              <mml:mo>)</mml:mo>\r\n                            </mml:mrow>\r\n                          </mml:mrow>\r\n                        </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>is considered in the strongly degenerate regime <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\ge 1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>p</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n                    <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>. The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\rightarrow \\infty \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:mrow>\r\n                            <mml:msup>\r\n                              <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n                                <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n                              </mml:mfrac>\r\n                            </mml:msup>\r\n                            <mml:msub>\r\n                              <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n                              <mml:mrow>\r\n                                <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:msup>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n                                    <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                              </mml:mrow>\r\n                            </mml:msub>\r\n                            <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n                            <mml:mspace />\r\n                            <mml:mtext>as</mml:mtext>\r\n                            <mml:mspace />\r\n                            <mml:mi>t</mml:mi>\r\n                            <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n                            <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n                        </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>The first result of this study complements this by asserting that given any positive <jats:inline-formula><jats:alternatives><jats:tex-math>$$f\\in C^0([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>f</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n                    <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n                      <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n                        <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n                        <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n                      </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> fulfilling <jats:inline-formula><jats:alternatives><jats:tex-math>$$f(t)\\rightarrow +\\infty $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>f</mml:mi>\r\n                    <mml:mo>(</mml:mo>\r\n                    <mml:mi>t</mml:mi>\r\n                    <mml:mo>)</mml:mo>\r\n                    <mml:mo>→</mml:mo>\r\n                    <mml:mo>+</mml:mo>\r\n                    <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> as <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\\rightarrow \\infty $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>t</mml:mi>\r\n                    <mml:mo>→</mml:mo>\r\n                    <mml:mi>∞</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> one can find a positive nondecreasing function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi \\in C^0([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>ϕ</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n                    <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n                      <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n                        <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n                        <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n                      </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> such that whenever <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0\\in C^0({\\mathbb {R}}^n)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n                    <mml:mo>∈</mml:mo>\r\n                    <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n                      <mml:msup>\r\n                        <mml:mrow>\r\n                          <mml:mi>R</mml:mi>\r\n                        </mml:mrow>\r\n                        <mml:mi>n</mml:mi>\r\n                      </mml:msup>\r\n                      <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> is radially symmetric with <jats:inline-formula><jats:alternatives><jats:tex-math>$$0&lt; u_0 &lt; \\phi (|\\cdot |)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mn>0</mml:mn>\r\n                    <mml:mo>&lt;</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n                    <mml:mrow>\r\n                      <mml:mo>&lt;</mml:mo>\r\n                      <mml:mi>ϕ</mml:mi>\r\n                      <mml:mo>(</mml:mo>\r\n                      <mml:mo>|</mml:mo>\r\n                    </mml:mrow>\r\n                    <mml:mo>·</mml:mo>\r\n                    <mml:mrow>\r\n                      <mml:mo>|</mml:mo>\r\n                      <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, the corresponding minimal solution <jats:italic>u</jats:italic> satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\frac{t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)}}{f(t)} \\rightarrow 0 \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:mrow>\r\n                            <mml:mfrac>\r\n                              <mml:mrow>\r\n                                <mml:msup>\r\n                                  <mml:mi>t</mml:mi>\r\n                                  <mml:mfrac>\r\n                                    <mml:mn>1</mml:mn>\r\n                                    <mml:mi>p</mml:mi>\r\n                                  </mml:mfrac>\r\n                                </mml:msup>\r\n                                <mml:msub>\r\n                                  <mml:mrow>\r\n                                    <mml:mo>‖</mml:mo>\r\n                                    <mml:mi>u</mml:mi>\r\n                                    <mml:mrow>\r\n                                      <mml:mo>(</mml:mo>\r\n                                      <mml:mo>·</mml:mo>\r\n                                      <mml:mo>,</mml:mo>\r\n                                      <mml:mi>t</mml:mi>\r\n                                      <mml:mo>)</mml:mo>\r\n                                    </mml:mrow>\r\n                                    <mml:mo>‖</mml:mo>\r\n                                  </mml:mrow>\r\n                                  <mml:mrow>\r\n                                    <mml:msup>\r\n                                      <mml:mi>L</mml:mi>\r\n                                      <mml:mi>∞</mml:mi>\r\n                                    </mml:msup>\r\n                                    <mml:mrow>\r\n                                      <mml:mo>(</mml:mo>\r\n                                      <mml:msup>\r\n                                        <mml:mrow>\r\n                                          <mml:mi>R</mml:mi>\r\n                                        </mml:mrow>\r\n                                        <mml:mi>n</mml:mi>\r\n                                      </mml:msup>\r\n                                      <mml:mo>)</mml:mo>\r\n                                    </mml:mrow>\r\n                                  </mml:mrow>\r\n                                </mml:msub>\r\n                              </mml:mrow>\r\n                              <mml:mrow>\r\n                                <mml:mi>f</mml:mi>\r\n                                <mml:mo>(</mml:mo>\r\n                                <mml:mi>t</mml:mi>\r\n                                <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n                            </mml:mfrac>\r\n                            <mml:mo>→</mml:mo>\r\n                            <mml:mn>0</mml:mn>\r\n                            <mml:mspace />\r\n                            <mml:mtext>as</mml:mtext>\r\n                            <mml:mspace />\r\n                            <mml:mi>t</mml:mi>\r\n                            <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n                            <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n                        </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>Secondly, (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n                    <mml:mi>u</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>. It is shown that if the connected components of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{u_0&gt;0\\}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mo>{</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n                    <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n                    <mml:mo>}</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>) satisfies <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} 0&lt; \\liminf _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} \\le \\limsup _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} &lt;\\infty . \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:mrow>\r\n                            <mml:mn>0</mml:mn>\r\n                            <mml:mo>&lt;</mml:mo>\r\n                            <mml:munder>\r\n                              <mml:mo>lim inf</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n                                <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n                              </mml:mrow>\r\n                            </mml:munder>\r\n                            <mml:mrow>\r\n                              <mml:mo>{</mml:mo>\r\n                            </mml:mrow>\r\n                            <mml:msup>\r\n                              <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n                                <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n                              </mml:mfrac>\r\n                            </mml:msup>\r\n                            <mml:msub>\r\n                              <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n                              <mml:mrow>\r\n                                <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:msup>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n                                    <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                              </mml:mrow>\r\n                            </mml:msub>\r\n                            <mml:mrow>\r\n                              <mml:mo>}</mml:mo>\r\n                            </mml:mrow>\r\n                            <mml:mo>≤</mml:mo>\r\n                            <mml:munder>\r\n                              <mml:mo>lim sup</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n                                <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n                              </mml:mrow>\r\n                            </mml:munder>\r\n                            <mml:mrow>\r\n                              <mml:mo>{</mml:mo>\r\n                            </mml:mrow>\r\n                            <mml:msup>\r\n                              <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n                                <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n                              </mml:mfrac>\r\n                            </mml:msup>\r\n                            <mml:msub>\r\n                              <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n                              <mml:mrow>\r\n                                <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n                                <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n                                  <mml:msup>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n                                    <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n                                  <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n                              </mml:mrow>\r\n                            </mml:msub>\r\n                            <mml:mrow>\r\n                              <mml:mo>}</mml:mo>\r\n                            </mml:mrow>\r\n                            <mml:mo>&lt;</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n                            <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n                        </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>Under a somewhat complementary hypothesis, particularly fulfilled if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{u_0&gt;0\\}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mo>{</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n                      <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n                    <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n                    <mml:mo>}</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.</jats:p>","lang":"eng"}],"user_id":"31496","_id":"53322","language":[{"iso":"eng"}],"keyword":["Analysis"],"issue":"S1","publication_status":"published","publication_identifier":{"issn":["1040-7294","1572-9222"]},"citation":{"apa":"Winkler, M. (2020). Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. <i>Journal of Dynamics and Differential Equations</i>, <i>36</i>(S1), 3–23. <a href=\"https://doi.org/10.1007/s10884-020-09892-x\">https://doi.org/10.1007/s10884-020-09892-x</a>","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.","bibtex":"@article{Winkler_2020, title={Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}, volume={36}, DOI={<a href=\"https://doi.org/10.1007/s10884-020-09892-x\">10.1007/s10884-020-09892-x</a>}, number={S1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={3–23} }","mla":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” <i>Journal of Dynamics and Differential Equations</i>, vol. 36, no. S1, Springer Science and Business Media LLC, 2020, pp. 3–23, doi:<a href=\"https://doi.org/10.1007/s10884-020-09892-x\">10.1007/s10884-020-09892-x</a>.","chicago":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” <i>Journal of Dynamics and Differential Equations</i> 36, no. S1 (2020): 3–23. <a href=\"https://doi.org/10.1007/s10884-020-09892-x\">https://doi.org/10.1007/s10884-020-09892-x</a>.","ieee":"M. Winkler, “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation,” <i>Journal of Dynamics and Differential Equations</i>, vol. 36, no. S1, pp. 3–23, 2020, doi: <a href=\"https://doi.org/10.1007/s10884-020-09892-x\">10.1007/s10884-020-09892-x</a>.","ama":"Winkler M. Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. <i>Journal of Dynamics and Differential Equations</i>. 2020;36(S1):3-23. doi:<a href=\"https://doi.org/10.1007/s10884-020-09892-x\">10.1007/s10884-020-09892-x</a>"},"page":"3-23","intvolume":"        36","year":"2020","author":[{"full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"date_created":"2024-04-07T12:37:38Z","volume":36,"publisher":"Springer Science and Business Media LLC","date_updated":"2024-04-07T12:37:45Z","doi":"10.1007/s10884-020-09892-x","title":"Approaching Critical Decay in a Strongly Degenerate Parabolic Equation"}]