On the Existence of Solutions of the Cauchy Problem for a Doubly Nonlinear Parabolic Equation

1996 ◽  
Vol 27 (5) ◽  
pp. 1235-1260 ◽  
Author(s):  
Kazuhiro Ishige
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Existence Of Solutions ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

scholarly journals The Decay Rate of the Solution to the Cauchy Problem for Doubly Nonlinear Parabolic Equation with Absortion

2020 ◽  
Author(s):  
З.В. Бесаева ◽  
А.Ф. Тедеев
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Decay Rate ◽  
Nonlinear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

В работе изучается задача Коши для широкого класса квазилнейных параболических уравнений второго порядка с неоднородной плотностью и абсорбцией. Хорошо известно, что для рассматриваемого класса задач без абсорбции и при условии, что плотность стремится к нулю не слишком быстро, имеет место закон сохранения тотальной массы. Однако этот факт не всегда имеет место при наличии абсорбции. В данной работе найдены точные условия на характер нелинейности и поведения неоднородной плотности на бесконечности, которые гарантируют стремление к нулю тотальной массы решения при неограниченном возрастании времени. Другими словами, найден критерий стабилизации к нулю тотальной массы решения в терминах критических показателей. С помощью полученных результатов и локальных оценок типа Нэша - Мозера выводятся точные оценки решения в равномерной метрике.

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

scholarly journals Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Hamid El Bahja ◽  
Abderrahmane El Hachimi ◽  
Ali Alami Idrissi
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Time Discretization ◽  
Global Compact ◽  
Compact Attractor ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.

ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION

1986 ◽  
Vol 54 (2) ◽  
pp. 421-455 ◽  
Author(s):  
V A Galaktionov ◽  
S P Kurdyumov ◽  
A A Samarskiĭ
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

scholarly journals Decay of solution of anisotropic doubly nonlinear parabolic equation in unbounded domains

2013 ◽  
Vol 5 (1) ◽  
pp. 63-82 ◽  
Author(s):  
Larisa Mikhailovna Kozhevnikova ◽  
Aleksei Aleksandrovich Leont'ev
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Unbounded Domains ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

scholarly journals Continuity of the temperature in a multi-phase transition problem

2021 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Phase Transition ◽  
Parabolic Equation ◽  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
Phase Transition Problem ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation ◽  
Multi Phase

AbstractLocally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the p-Laplacian type diffusion is also considered.

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>

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