On a degenerate parabolic equation from double phase convection
Abstract The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a ( x ) $a(x)$ and b ( x ) $b(x)$ be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a ( x ) + b ( x ) >...
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2021
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oai:doaj.org-article:cb71353c66194d97a28ad3864e422a6e2021-11-28T12:08:21ZOn a degenerate parabolic equation from double phase convection10.1186/s13662-021-03659-41687-1847https://doaj.org/article/cb71353c66194d97a28ad3864e422a6e2021-11-01T00:00:00Zhttps://doi.org/10.1186/s13662-021-03659-4https://doaj.org/toc/1687-1847Abstract The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a ( x ) $a(x)$ and b ( x ) $b(x)$ be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a ( x ) + b ( x ) > 0 $a(x)+b(x)>0$ , x ∈ Ω ‾ $x\in \overline{\Omega }$ and the boundary value condition should be imposed. In this paper, the condition a ( x ) + b ( x ) > 0 $a(x)+b(x)>0$ , x ∈ Ω ‾ $x\in \overline{\Omega }$ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and u t ∈ L 2 ( Q T ) $u_{t}\in L^{2}(Q_{T})$ is shown. The stability of weak solutions is studied according to the different integrable conditions of a ( x ) $a(x)$ and b ( x ) $b(x)$ . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by a ( x ) b ( x ) | x ∈ ∂ Ω = 0 $a(x)b(x)|_{x\in \partial \Omega }=0$ is found for the first time.Huashui ZhanSpringerOpenarticleDegenerate parabolic equationDouble phase convectionBoundary value conditionTraceMathematicsQA1-939ENAdvances in Difference Equations, Vol 2021, Iss 1, Pp 1-32 (2021) |
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DOAJ |
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DOAJ |
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Degenerate parabolic equation Double phase convection Boundary value condition Trace Mathematics QA1-939 |
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Degenerate parabolic equation Double phase convection Boundary value condition Trace Mathematics QA1-939 Huashui Zhan On a degenerate parabolic equation from double phase convection |
| description |
Abstract The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a ( x ) $a(x)$ and b ( x ) $b(x)$ be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a ( x ) + b ( x ) > 0 $a(x)+b(x)>0$ , x ∈ Ω ‾ $x\in \overline{\Omega }$ and the boundary value condition should be imposed. In this paper, the condition a ( x ) + b ( x ) > 0 $a(x)+b(x)>0$ , x ∈ Ω ‾ $x\in \overline{\Omega }$ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and u t ∈ L 2 ( Q T ) $u_{t}\in L^{2}(Q_{T})$ is shown. The stability of weak solutions is studied according to the different integrable conditions of a ( x ) $a(x)$ and b ( x ) $b(x)$ . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by a ( x ) b ( x ) | x ∈ ∂ Ω = 0 $a(x)b(x)|_{x\in \partial \Omega }=0$ is found for the first time. |
| format |
article |
| author |
Huashui Zhan |
| author_facet |
Huashui Zhan |
| author_sort |
Huashui Zhan |
| title |
On a degenerate parabolic equation from double phase convection |
| title_short |
On a degenerate parabolic equation from double phase convection |
| title_full |
On a degenerate parabolic equation from double phase convection |
| title_fullStr |
On a degenerate parabolic equation from double phase convection |
| title_full_unstemmed |
On a degenerate parabolic equation from double phase convection |
| title_sort |
on a degenerate parabolic equation from double phase convection |
| publisher |
SpringerOpen |
| publishDate |
2021 |
| url |
https://doaj.org/article/cb71353c66194d97a28ad3864e422a6e |
| work_keys_str_mv |
AT huashuizhan onadegenerateparabolicequationfromdoublephaseconvection |
| _version_ |
1718408225418641408 |