| Summary: | In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close=""><mtable><mtr><mtd columnalign="left"><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>+</mo><mi>M</mi><mrow><mo>(</mo><msubsup><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mi>s</mi><mn>2</mn></msubsup><mo>)</mo></mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mi>ln</mi><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mo>,</mo><mspace width="4pt"></mspace></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>in</mi></mrow><mo> </mo><mo>Ω</mo><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>in</mi></mrow><mo> </mo><mo>Ω</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mi>on</mi></mrow><mo> </mo><mi mathvariant="sans-serif-italic">∂</mi><mo>Ω</mo><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mfenced></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mi>s</mi></msub></semantics></math></inline-formula> is the Gagliardo semi-norm of <i>u</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup></semantics></math></inline-formula> is the fractional Laplacian, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>λ</mi><mo><</mo><mi>p</mi><mo><</mo><msubsup><mn>2</mn><mi>s</mi><mo>*</mo></msubsup><mo>=</mo><mn>2</mn><mi>N</mi><mo>/</mo><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow></semantics></math></inline-formula> is a bounded domain with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>></mo><mn>2</mn><mi>s</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>u</mi><mn>0</mn></msub></semantics></math></inline-formula> is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level.
|
|---|