scholarly journals Blowup of Solution for a Class of Doubly Nonlinear Parabolic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jun Lu ◽  
Qingying Hu ◽  
Hongwei Zhang
Keyword(s):  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Blowup Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Initial Boundary Value ◽  
Suitable Initial Data

An initial boundary value problem for a class of doubly parabolic equations is studied. We obtain sufficient conditions for the blowup of solutions under suitable initial data using differential inequalities.

A NONLINEAR PARABOLIC SYSTEM ARISING IN THERMOMECHANICS AND IN THERMOMAGNETISM

1992 ◽  
Vol 02 (03) ◽  
pp. 271-281 ◽  
Author(s):  
JOSÉ-FRANCISCO RODRIGUES
Keyword(s):  
Parabolic Equations ◽  
Eddy Currents ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Arbitrary Cross Section ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value ◽  
Dependent Viscosity

We consider a system of two parabolic equations modeling the thermo-convection of a Newtonian fluid, with temperature dependent viscosity of energy dissipation, as well as the thermal effects of the eddy currents, induced by a slowly varying magnetic field, in cylinders with arbitrary cross-section. We show the existence of a weak solution of the corresponding initial-boundary value problem and, under additional assumptions, we consider the question of the uniqueness and regularity of the solution.

STRONG SOLUTIONS FOR STOCHASTIC NONLINEAR NONLOCAL PARABOLIC EQUATIONS

2010 ◽  
Vol 10 (04) ◽  
pp. 497-508 ◽  
Author(s):  
EDSON A. COAYLA-TERAN
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Multiplicative White Noise ◽  
Nonlocal Parabolic Equations ◽  
Initial Boundary Value

In this article we investigate the existence and uniqueness of strong solutions to the initial-boundary value problem with homogeneous boundary conditions for a stochastic nonlinear parabolic equation of nonlocal type with multiplicative white noise. Moreover, we prove a simple result on the asymptotic behavior for the solution.

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>

XXII.—A Priori Estimates and Nonlinear Parabolic Equations of Arbitrary Order

1972 ◽  
Vol 69 (4) ◽  
pp. 287-293
Author(s):  
D. E. Edmunds ◽  
C. A. Stuart
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Initial Boundary Value

SynopsisIn this paper it is shown that the question of the existence of a classical solution of the first initial-boundary value problem for a non-linear parabolic equation may be reduced to the problem of the derivation of suitable a priori bounds.

scholarly journals The weak solutions of a doubly nonlinear parabolic equation related to the $p(x)$-Laplacian

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation ◽  
The Stability ◽  
Initial Boundary Value

Abstract A nonlinear degenerate parabolic equation related to the $p(x)$p(x)-Laplacian $$ {u_{t}}= \operatorname{div} \bigl({b(x)} { \bigl\vert {\nabla a(u)} \bigr\vert ^{p(x) - 2}}\nabla a(u) \bigr)+\sum _{i=1}^{N}\frac{\partial b_{i}(u)}{ \partial x_{i}}+c(x,t) -b_{0}a(u) $$ut=div(b(x)|∇a(u)|p(x)−2∇a(u))+∑i=1N∂bi(u)∂xi+c(x,t)−b0a(u) is considered in this paper, where $b(x)|_{x\in \varOmega }>0$b(x)|x∈Ω>0, $b(x)|_{x \in \partial \varOmega }=0$b(x)|x∈∂Ω=0, $a(s)\geq 0$a(s)≥0 is a strictly increasing function with $a(0)=0$a(0)=0, $c(x,t)\geq 0$c(x,t)≥0 and $b_{0}>0$b0>0. If $\int _{\varOmega }b(x)^{-\frac{1}{p ^{-}-1}}\,dx\leq c$∫Ωb(x)−1p−−1dx≤c and $\vert \sum_{i=1}^{N}b_{i}'(s) \vert \leq c a'(s)$|∑i=1Nbi′(s)|≤ca′(s), then the solutions of the initial-boundary value problem is well-posedness. When $\int _{\varOmega }b(x)^{-(p(x)-1)}\,dx<\infty $∫Ωb(x)−(p(x)−1)dx<∞, without the boundary value condition, the stability of weak solutions can be proved.

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