Implicit difference method for solving the first boundary-value problem for a second-order nonlinear parabolic equation

1977 ◽  
Vol 7 (1) ◽  
pp. 123-127
Author(s):  
M. N. Yakovlev
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Nonlinear Parabolic Equation ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Parabolic

scholarly journals Solvability of doubly nonlinear parabolic equation with p-laplacian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shun Uchida
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Nonlinear Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we consider a doubly nonlinear parabolic equation <inline-formula><tex-math id="M2">\begin{document}$ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $\end{document}</tex-math></inline-formula> with the homogeneous Dirichlet boundary condition in a bounded domain, where <inline-formula><tex-math id="M3">\begin{document}$ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $\end{document}</tex-math></inline-formula> is a maximal monotone graph satisfying <inline-formula><tex-math id="M4">\begin{document}$ 0 \in \beta (0) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \nabla \cdot \alpha (x , \nabla u ) $\end{document}</tex-math></inline-formula> stands for a generalized <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on <inline-formula><tex-math id="M7">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for <inline-formula><tex-math id="M8">\begin{document}$ 1 &lt; p &lt; 2 $\end{document}</tex-math></inline-formula>. Main purpose of this paper is to show the solvability of the initial boundary value problem for any <inline-formula><tex-math id="M9">\begin{document}$ p \in (1, \infty ) $\end{document}</tex-math></inline-formula> without any conditions for <inline-formula><tex-math id="M10">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> except <inline-formula><tex-math id="M11">\begin{document}$ 0 \in \beta (0) $\end{document}</tex-math></inline-formula>. We also discuss the uniqueness of solution by using properties of entropy solution.</p>

scholarly journals Boundary layer study of a nonlinear parabolic equation with a small parameter

2021 ◽  
Author(s):  
qin xulong ◽  
xu zhao ◽  
wenshu zhou
Keyword(s):  
Boundary Layer ◽  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Nonlinear Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation with a small parameter. The existence of a boundary layer as the parameter goes to zero is obtained together with the estimation on the thickness of the boundary layer. The main result extends an earlier work of Frid and Shelukhin (1999).

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