scholarly journals Fujita type global solvability to double nonlinear parabolic equation with nonlinear gradient term for the salt transfer process

2021 ◽  
Vol 2131 (3) ◽  
pp. 032042
Author(s):  
M M Aripov ◽  
R B Baltabaeva
Keyword(s):  
Parabolic Equation ◽  
Global Solvability ◽  
Gradient Term ◽  
Fast Diffusion ◽  
Qualitative Properties ◽  
Splitting Algorithm ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem ◽  
Double Nonlinear ◽  
Degenerate Type

Abstract In this paper we investigated the qualitative properties of non-negative and bounded continuous solutions to problem Cauchy for a degenerate parabolic equation with nonlinear gradient term. We based on splitting algorithm suggest estimate of weak solution for slowly, fast diffusion and critical cases, Fujita type global solvability of the Cauchy problem to degenerate type parabolic equation with nonlinear gradient term established. The theorems proven with comparison principle.

Global existence and non-existence for the degenerate and uniformly parabolic equations with gradient term

2013 ◽  
Vol 143 (3) ◽  
pp. 643-668 ◽  
Author(s):  
Haifeng Shang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Global Solvability ◽  
Gradient Term ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Limiting Case

We study the Cauchy problem for the degenerate and uniformly parabolic equations with gradient term. The local existence, global existence and non-existence of solutions are obtained. In the case of global solvability, we get the exact estimates of a solution. In particular, we obtain the global existence of solutions in the limiting case.

scholarly journals Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density

2021 ◽  
Vol 2131 (3) ◽  
pp. 032043
Author(s):  
M Aripov ◽  
A S Matyakubov ◽  
J O Khasanov ◽  
M M Bobokandov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Variable Density ◽  
Divergence Form ◽  
Cross Diffusion ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

Abstract In this paper the properties of solutions of nonlinear parabolic equation not in divergence form | x | − 1 ∂ u ∂ t = u q ∂ ∂ x ( | x | n u m − 1 | ∂ u k ∂ x | p − 2 ∂ u ∂ x ) + | x | − 1 u β are studied. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic equation not in divergence form, depending on the value in the equation of the numerical parameters necessary and sufficient signs of their existence. The compactly supported solution of the Cauchy problem for a cross-diffusion parabolic equation not in divergence form with a source and a variable density is obtained.

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.

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