Continuity of the temperature in a multi-phase transition problem

Locally 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.

Intrinsic scaling method for doubly nonlinear parabolic equations and its application

In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

Finite extinction for a doubly nonlinear parabolic equation of fast diffusion type

PurposeThe purpose of this paper is to find a doubly nonlinear parabolic equation of fast diffusion in a bounded domain.Design/methodology/approachFor positive and bounded initial data, the authors study the initial zero-boundary value problem.FindingsThe findings of this study showed the complete extinction of a continuous weak solution at a finite time.Originality/valueThe extinction time is studied earlier but for the Laplacian case. The authors presented the finite extinction time for the case of p-Laplacian.

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

In 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.

Solvability of doubly nonlinear parabolic equation with p-laplacian

<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>

Closed loop control of gas flow in a pipe: stability for a transient model

This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=\frac{3}{2}. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow.We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an {L^{2}}-sense. The analysis is done both for the PDE system and an ODE system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.

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

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.

Regularity Estimates for the p-Sobolev Flow

We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case $$p=2$$p=2. In this article we establish a priori estimates and regularity results for the p-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.

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

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.