On a degenerate parabolic equation from double phase convection

Media type: E-Article
Title: On a degenerate parabolic equation from double phase convection
Contributor: Zhan, Huashui
imprint: Springer Science and Business Media LLC, 2021
Published in: Advances in Difference Equations
Language: English
DOI: 10.1186/s13662-021-03659-4
ISSN: 1687-1847
Keywords: Applied Mathematics ; Algebra and Number Theory ; Analysis
Origination:
Footnote:
Description: <jats:title>Abstract</jats:title><jats:p>The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let <jats:inline-formula><jats:alternatives><jats:tex-math>$a(x)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$b(x)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math></jats:alternatives></jats:inline-formula> be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that <jats:inline-formula><jats:alternatives><jats:tex-math>$a(x)+b(x)>0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$x\in \overline{\Omega }$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>‾</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula> and the boundary value condition should be imposed. In this paper, the condition <jats:inline-formula><jats:alternatives><jats:tex-math>$a(x)+b(x)>0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$x\in \overline{\Omega }$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>‾</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula> is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution <jats:italic>u</jats:italic> is proved by parabolically regularized method, and <jats:inline-formula><jats:alternatives><jats:tex-math>$u_{t}\in L^{2}(Q_{T})$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math></jats:alternatives></jats:inline-formula> is shown. The stability of weak solutions is studied according to the different integrable conditions of <jats:inline-formula><jats:alternatives><jats:tex-math>$a(x)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$b(x)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math></jats:alternatives></jats:inline-formula>. To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by <jats:inline-formula><jats:alternatives><jats:tex-math>$a(x)b(x)|_{x\in \partial \Omega }=0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:msub> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math></jats:alternatives></jats:inline-formula> is found for the first time.</jats:p>
Access State: Open Access

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