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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>.","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>","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} }","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>","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>.","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>.","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23."},"language":[{"iso":"eng"}],"doi":"10.1007/s10884-020-09892-x","publication_identifier":{"issn":["1040-7294","1572-9222"]},"author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"year":"2020","title":"Approaching Critical Decay in a Strongly Degenerate Parabolic Equation","intvolume":"        36","publication_status":"published","date_updated":"2024-04-07T12:37:45Z","date_created":"2024-04-07T12:37:38Z","type":"journal_article","keyword":["Analysis"],"issue":"S1","publication":"Journal of Dynamics and Differential Equations","abstract":[{"lang":"eng","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>"}]},{"status":"public","user_id":"70575","volume":378,"page":"917-941","publisher":"Springer Science and Business Media LLC","_id":"53415","citation":{"ama":"Küster B, Weich T. Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>. 2020;378(2):917-941. doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>","bibtex":"@article{Küster_Weich_2020, title={Pollicott-Ruelle Resonant States and Betti Numbers}, volume={378}, DOI={<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>}, number={2}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Küster, Benjamin and Weich, Tobias}, year={2020}, pages={917–941} }","mla":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, Springer Science and Business Media LLC, 2020, pp. 917–41, doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>.","chicago":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i> 378, no. 2 (2020): 917–41. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>.","short":"B. Küster, T. Weich, Communications in Mathematical Physics 378 (2020) 917–941.","apa":"Küster, B., &#38; Weich, T. (2020). Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>, <i>378</i>(2), 917–941. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>","ieee":"B. Küster and T. Weich, “Pollicott-Ruelle Resonant States and Betti Numbers,” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, pp. 917–941, 2020, doi: <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>."},"publication_status":"published","date_updated":"2024-04-11T12:36:53Z","intvolume":"       378","title":"Pollicott-Ruelle Resonant States and Betti Numbers","year":"2020","publication_identifier":{"issn":["0010-3616","1432-0916"]},"author":[{"full_name":"Küster, Benjamin","first_name":"Benjamin","last_name":"Küster"},{"full_name":"Weich, Tobias","orcid":"0000-0002-9648-6919","last_name":"Weich","first_name":"Tobias","id":"49178"}],"doi":"10.1007/s00220-020-03793-2","language":[{"iso":"eng"}],"abstract":[{"text":"<jats:title>Abstract</jats:title><jats:p>Given a closed orientable hyperbolic manifold of dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ne 3$$</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:mn>3</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.\r\n</jats:p>","lang":"eng"}],"issue":"2","publication":"Communications in Mathematical Physics","type":"journal_article","keyword":["Mathematical Physics","Statistical and Nonlinear Physics"],"department":[{"_id":"548"}],"date_created":"2024-04-11T12:33:03Z"},{"publication_identifier":{"eissn":["2625-0675"]},"author":[{"last_name":"Pieper","first_name":"Catania","full_name":"Pieper, Catania"},{"first_name":"Brigitte","last_name":"Kottmann","full_name":"Kottmann, Brigitte","id":"82466"},{"last_name":"Miller","first_name":"Susanne","full_name":"Miller, Susanne"}],"title":"Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung","year":"2020","intvolume":"         3","date_updated":"2024-04-11T17:13:34Z","publication_status":"published","language":[{"iso":"ger"}],"doi":"10.4119/HLZ-2484","publication":"HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion","issue":"1","abstract":[{"lang":"ger","text":"In dem vorliegenden Beitrag werden insgesamt zehn authentische Fallportraits von Bielefelder Grundschulkindern vorgestellt. Sie bieten sowohl Einblicke in die familiäre als auch in die schulische Situation von Kindern, so dass heterogene Lebenswirklichkeiten mehrperspektivisch abgebildet werden. Die Fallportraits können im Rahmen der phasenübergreifenden Lehrer_innenbilung eingesetzt werden, um bspw. Transferprozesse zwischen Theorie und Praxis zu unterstützen, eine kritisch-reflexive Auseinandersetzung sowie ein Sensibilisieren für heterogene Lebenswirklichkeiten zu fördern, etc. Didaktische Implementierungsmöglichkeiten und Anknüpfungspunkte zur Portfolioarbeit (2), vier exemplarische theoretische Schwerpunkte (3) sowie der Prozess der Fallportraitgenerierung (4) werden zunächst im vorliegenden Beitrag erläutert, ehe die zehn Fallportraits (5), ein Fazit (6) und zwei exemplarische Veranschaulichungsmaterialien (7) den Beitrag beenden."}],"date_created":"2023-01-25T07:39:35Z","file":[{"date_created":"2023-01-25T07:44:31Z","creator":"schnelll","file_id":"39822","content_type":"application/pdf","success":1,"file_name":"2484-Artikeltext-16104-1-10-20200204 2.pdf","file_size":3057643,"access_level":"closed","relation":"main_file","date_updated":"2023-01-25T07:44:31Z"}],"department":[{"_id":"479"}],"type":"journal_article","status":"public","has_accepted_license":"1","_id":"39821","page":"94-171","volume":3,"ddc":["370"],"user_id":"71040","citation":{"bibtex":"@article{Pieper_Kottmann_Miller_2020, title={Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung}, volume={3}, DOI={<a href=\"https://doi.org/10.4119/HLZ-2484\">10.4119/HLZ-2484</a>}, number={1}, journal={HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion}, author={Pieper, Catania and Kottmann, Brigitte and Miller, Susanne}, year={2020}, pages={94–171} }","ama":"Pieper C, Kottmann B, Miller S. Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung. <i>HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion</i>. 2020;3(1):94-171. doi:<a href=\"https://doi.org/10.4119/HLZ-2484\">10.4119/HLZ-2484</a>","mla":"Pieper, Catania, et al. “Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung.” <i>HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion</i>, vol. 3, no. 1, 2020, pp. 94–171, doi:<a href=\"https://doi.org/10.4119/HLZ-2484\">10.4119/HLZ-2484</a>.","chicago":"Pieper, Catania, Brigitte Kottmann, and Susanne Miller. “Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung.” <i>HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion</i> 3, no. 1 (2020): 94–171. <a href=\"https://doi.org/10.4119/HLZ-2484\">https://doi.org/10.4119/HLZ-2484</a>.","short":"C. Pieper, B. Kottmann, S. Miller, HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion 3 (2020) 94–171.","ieee":"C. Pieper, B. Kottmann, and S. Miller, “Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung,” <i>HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion</i>, vol. 3, no. 1, pp. 94–171, 2020, doi: <a href=\"https://doi.org/10.4119/HLZ-2484\">10.4119/HLZ-2484</a>.","apa":"Pieper, C., Kottmann, B., &#38; Miller, S. (2020). Fallportraits von Bielefelder Grundschulkindern. Fallarbeit in der (phasenübergreifenden) Lehrer_innenbildung. <i>HLZ - Herausforderung Lehrer_innenbildung, Zeitschrift zur Konzeption, Gestaltung und Diskussion</i>, <i>3</i>(1), 94–171. <a href=\"https://doi.org/10.4119/HLZ-2484\">https://doi.org/10.4119/HLZ-2484</a>"},"file_date_updated":"2023-01-25T07:44:31Z"},{"type":"conference","department":[{"_id":"36"}],"file":[{"content_type":"application/pdf","success":1,"file_id":"53478","date_updated":"2024-04-14T12:02:00Z","relation":"main_file","access_level":"closed","file_size":283168,"file_name":"BzMu2020_MALIK-id259.pdf","date_created":"2024-04-14T12:02:00Z","creator":"smalik"}],"date_created":"2024-04-14T12:02:24Z","place":"Münster","file_date_updated":"2024-04-14T12:02:00Z","publication":"Beiträge zum Mathematikunterricht 2020. 54. Jahrestagung der Gesellschaft für Didaktik der Mathematik vom 09. bis 13. März 2020 in Würzburg","citation":{"short":"S.N. Malik, in: Beiträge Zum Mathematikunterricht 2020. 54. Jahrestagung Der Gesellschaft Für Didaktik Der Mathematik Vom 09. Bis 13. März 2020 in Würzburg, WTM-Verlag, Münster, 2020.","chicago":"Malik, Sara Naseem. “Die Curriculare Entwicklung Der Anforderungen von Anwendungsbezogenen Aufgaben.” In <i>Beiträge Zum Mathematikunterricht 2020. 54. Jahrestagung Der Gesellschaft Für Didaktik Der Mathematik Vom 09. Bis 13. März 2020 in Würzburg</i>. Münster: WTM-Verlag, 2020. <a href=\"https://doi.org/10.17877/DE290R-21455\">https://doi.org/10.17877/DE290R-21455</a>.","ieee":"S. N. Malik, “Die curriculare Entwicklung der Anforderungen von anwendungsbezogenen Aufgaben,” 2020, doi: <a href=\"https://doi.org/10.17877/DE290R-21455\">10.17877/DE290R-21455</a>.","apa":"Malik, S. N. (2020). Die curriculare Entwicklung der Anforderungen von anwendungsbezogenen Aufgaben. <i>Beiträge Zum Mathematikunterricht 2020. 54. Jahrestagung Der Gesellschaft Für Didaktik Der Mathematik Vom 09. Bis 13. März 2020 in Würzburg</i>. <a href=\"https://doi.org/10.17877/DE290R-21455\">https://doi.org/10.17877/DE290R-21455</a>","bibtex":"@inproceedings{Malik_2020, place={Münster}, title={Die curriculare Entwicklung der Anforderungen von anwendungsbezogenen Aufgaben}, DOI={<a href=\"https://doi.org/10.17877/DE290R-21455\">10.17877/DE290R-21455</a>}, booktitle={Beiträge zum Mathematikunterricht 2020. 54. Jahrestagung der Gesellschaft für Didaktik der Mathematik vom 09. bis 13. März 2020 in Würzburg}, publisher={WTM-Verlag}, author={Malik, Sara Naseem}, year={2020} }","ama":"Malik SN. Die curriculare Entwicklung der Anforderungen von anwendungsbezogenen Aufgaben. In: <i>Beiträge Zum Mathematikunterricht 2020. 54. Jahrestagung Der Gesellschaft Für Didaktik Der Mathematik Vom 09. Bis 13. März 2020 in Würzburg</i>. WTM-Verlag; 2020. doi:<a href=\"https://doi.org/10.17877/DE290R-21455\">10.17877/DE290R-21455</a>","mla":"Malik, Sara Naseem. “Die Curriculare Entwicklung Der Anforderungen von Anwendungsbezogenen Aufgaben.” <i>Beiträge Zum Mathematikunterricht 2020. 54. Jahrestagung Der Gesellschaft Für Didaktik Der Mathematik Vom 09. Bis 13. März 2020 in Würzburg</i>, WTM-Verlag, 2020, doi:<a href=\"https://doi.org/10.17877/DE290R-21455\">10.17877/DE290R-21455</a>."},"user_id":"38640","doi":"10.17877/DE290R-21455","ddc":["370","510"],"_id":"53477","language":[{"iso":"eng"}],"publisher":"WTM-Verlag","date_updated":"2024-04-14T12:02:34Z","has_accepted_license":"1","year":"2020","status":"public","title":"Die curriculare Entwicklung der Anforderungen von anwendungsbezogenen Aufgaben","author":[{"id":"38640","last_name":"Malik","first_name":"Sara Naseem","full_name":"Malik, Sara Naseem"}]},{"publication":" MusikTexte","citation":{"chicago":"Ricke, Anna. “Im Schweigen der Klänge. Zu Farzia Fallahs Sextett ‘im selben augenblick’ (2018).” <i> MusikTexte</i> 165 (2020): 11–14.","ama":"Ricke A. Im Schweigen der Klänge. Zu Farzia Fallahs Sextett “im selben augenblick” (2018). <i> MusikTexte</i>. 2020;165:11-14.","short":"A. Ricke,  MusikTexte 165 (2020) 11–14.","bibtex":"@article{Ricke_2020, title={Im Schweigen der Klänge. Zu Farzia Fallahs Sextett “im selben augenblick” (2018)}, volume={165}, journal={ MusikTexte}, author={Ricke, Anna}, year={2020}, pages={11–14} }","apa":"Ricke, A. (2020). Im Schweigen der Klänge. Zu Farzia Fallahs Sextett “im selben augenblick” (2018). <i> MusikTexte</i>, <i>165</i>, 11–14.","mla":"Ricke, Anna. “Im Schweigen der Klänge. Zu Farzia Fallahs Sextett ‘im selben augenblick’ (2018).” <i> MusikTexte</i>, vol. 165, 2020, pp. 11–14.","ieee":"A. Ricke, “Im Schweigen der Klänge. Zu Farzia Fallahs Sextett ‘im selben augenblick’ (2018),” <i> MusikTexte</i>, vol. 165, pp. 11–14, 2020."},"type":"journal_article","date_created":"2024-04-15T12:02:07Z","publication_status":"published","date_updated":"2024-04-15T12:16:26Z","intvolume":"       165","status":"public","title":"Im Schweigen der Klänge. Zu Farzia Fallahs Sextett \"im selben augenblick\" (2018)","year":"2020","author":[{"full_name":"Ricke, Anna","first_name":"Anna","last_name":"Ricke","id":"78162"}],"user_id":"78162","volume":165,"page":"11-14","_id":"53511","language":[{"iso":"ger"}]},{"date_created":"2020-08-01T07:20:33Z","keyword":["Securitization","Credit risk transfer","Effective tax rates","European banking"],"type":"journal_article","department":[{"_id":"19"}],"publication":"The Quarterly Review of Economics and Finance","citation":{"ieee":"A. Uhde, “Tax avoidance through securitization,” <i>The Quarterly Review of Economics and Finance</i>, 2020, doi: <a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">10.1016/j.qref.2020.07.008</a>.","apa":"Uhde, A. (2020). Tax avoidance through securitization. <i>The Quarterly Review of Economics and Finance</i>. <a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">https://doi.org/10.1016/j.qref.2020.07.008</a>","chicago":"Uhde, André. “Tax Avoidance through Securitization.” <i>The Quarterly Review of Economics and Finance</i>, 2020. <a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">https://doi.org/10.1016/j.qref.2020.07.008</a>.","short":"A. Uhde, The Quarterly Review of Economics and Finance (2020).","mla":"Uhde, André. “Tax Avoidance through Securitization.” <i>The Quarterly Review of Economics and Finance</i>, 2020, doi:<a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">10.1016/j.qref.2020.07.008</a>.","bibtex":"@article{Uhde_2020, title={Tax avoidance through securitization}, DOI={<a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">10.1016/j.qref.2020.07.008</a>}, journal={The Quarterly Review of Economics and Finance}, author={Uhde, André}, year={2020} }","ama":"Uhde A. Tax avoidance through securitization. <i>The Quarterly Review of Economics and Finance</i>. Published online 2020. doi:<a href=\"https://doi.org/10.1016/j.qref.2020.07.008\">10.1016/j.qref.2020.07.008</a>"},"abstract":[{"lang":"eng","text":"Employing a unique hand-collected sample of 956 credit risk securitization transactions issued by 64 stock-listed European banks across the EU-13 plus Switzerland over the period from 1997 to 2010, this paper empirically analyzes the impact of securitization on the issuing banks’ effective tax rates. Our analysis reveals that banks may reduce their tax expense through securitization via a direct and indirect channel suggesting that tax avoidance may be a further motive for banks to engage in the securitization business. These baseline findings remain robust under various robustness checks, especially when implementing structural equation models and controlling for a reverse causality between the banks’ tax burden and their incentive to securitize. Finally, various sensitivity analyses provide further important results and implications for tax policies, banking regulation and the ongoing process of revitalizing the European securitization market."}],"language":[{"iso":"eng"}],"_id":"17522","doi":"10.1016/j.qref.2020.07.008","user_id":"36049","status":"public","year":"2020","title":"Tax avoidance through securitization","author":[{"full_name":"Uhde, André","first_name":"André","last_name":"Uhde","orcid":"https://orcid.org/0000-0002-8058-8857","id":"36049"}],"publication_identifier":{"issn":["1062-9769"]},"jel":["G21","G28","H25","H71"],"date_updated":"2024-04-17T13:35:56Z","article_type":"original"},{"citation":{"ieee":"A. Uhde, “Tax avoidance through securitization,” <i>The Quarterly Review of Economics and Finance</i>, 2020.","apa":"Uhde, A. (2020). Tax avoidance through securitization. <i>The Quarterly Review of Economics and Finance</i>.","chicago":"Uhde, André. “Tax Avoidance through Securitization.” <i>The Quarterly Review of Economics and Finance</i>, 2020.","short":"A. Uhde, The Quarterly Review of Economics and Finance (2020).","mla":"Uhde, André. “Tax Avoidance through Securitization.” <i>The Quarterly Review of Economics and Finance</i>, 2020.","bibtex":"@article{Uhde_2020, title={Tax avoidance through securitization}, journal={The Quarterly Review of Economics and Finance}, author={Uhde, André}, year={2020} }","ama":"Uhde A. Tax avoidance through securitization. <i>The Quarterly Review of Economics and Finance</i>. Published online 2020."},"publication":"The Quarterly Review of Economics and Finance","abstract":[{"text":"Employing a unique hand-collected sample of 956 credit risk securitization transactions issued by 64 stock-listed European banks across the EU-13 plus Switzerland over the period from 1997 to 2010, this paper empirically analyzes the impact of securitization on the issuing banks’ effective tax rates. Our analysis reveals that banks may reduce their tax expense through securitization via a direct and indirect channel suggesting that tax avoidance may be a further motive for banks to engage in the securitization business. These baseline findings remain robust under various robustness checks, especially when implementing structural equation models and controlling for a reverse causality between the banks’ tax burden and their incentive to securitize. Finally, various sensitivity analyses provide further important results and implications for tax policies, banking regulation and the ongoing process of revitalizing the European securitization market.","lang":"eng"}],"date_created":"2020-07-20T06:29:36Z","department":[{"_id":"19"}],"type":"journal_article","keyword":["Securitization","credit risk transfer","effective tax rates","European banking"],"author":[{"full_name":"Uhde, André","first_name":"André","last_name":"Uhde","orcid":"https://orcid.org/0000-0002-8058-8857","id":"36049"}],"jel":["G21","G28","H25","H71"],"title":"Tax avoidance through securitization","year":"2020","status":"public","date_updated":"2024-04-17T13:36:03Z","_id":"17401","language":[{"iso":"eng"}],"user_id":"36049"}]
