A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications
type: Item
instance of: scholarly article
title: A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications
author: Yu Ichida
published in: Communications on Pure and Applied Analysis
publication date: 2024-08-13T00:00:00Z
review text: The paper examines two parabolic differential equations. The first is the spatial one-dimensional degenerate equation \N\[\NU_t=U^p(U_{xx}+\mu U)-\delta U,\;\;t>0,\;\;x\in\mathbf{R},\N\]\Nwhere \(p>0, \mu>0,\) and \(\delta=0\) or \(1.\) The second is \N\[\NV_t=(V^m)_{xx}+\mu V^m-\delta(1-p)V,\;\;t>0,\;\;x\in{\mathbf{R}}, \;\;m>1,\N\]\Nobtained from the first one by the transformation \N\[\NU(t, x)=(1-p)^{-\frac{1}{p}}(V(t,x))^{\frac{1}{1-p}},\N\]\Nwhere \(0<p<1.\)\N\NThe author point out in the abstract: ``The characterization of stationary solutions given in this paper refers to the enumeration of those that exist and the presentation of solution information such as the profile and asymptotic behavior of each of them. Due to the influence of terms derived from the degeneracy of the equations, it is not easy to classify and characterize the existence of stationary solutions of the equations considered in this paper. These results are obtained by using dynamical systems theory and geometric approaches (in particular, Poincaré compactification). In addition, an application of the results obtained in this paper is given. The results of the characterization of nonnegative weak stationary solutions of the spatial one-dimensional porous medium equation with special nonlinear terms are shown. These can be obtained by carefully using transformations known from previous studies.''
reviewed by: Petio S. Kelevedjiev
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 7896652
zbMATH Keywords: 1D degenerate parabolic equation
zbMATH Keywords: 1D porous medium equation
zbMATH Keywords: Poincaré compactification
zbMATH Keywords: nonnegative stationary solution
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
cites work: Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation
cites work: Asymptotic expansions of traveling wave solutions for a quasilinear parabolic equation
cites work: On the formation of singularities in the curve shortening flow
cites work: Asymptotic shape of cusp singularities in curve shortening
cites work: Qualitative theory of planar differential systems
cites work: Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity
cites work: A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations
cites work: Traveling wave solutions for degenerate nonlinear parabolic equations
cites work: Stationary solutions for a 1D PDE problem with gradient term and negative powers nonlinearity
cites work: Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations
cites work: Multiple time scale dynamics
cites work: Contracting convex immersed closed plane curves with slow speed of curvature
cites work: On Blow-Up Solutions of Differential Equations with Poincaré-Type Compactifications
cites work: Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs
cites work: Blowup rate of solutions of a degenerate nonlinear parabolic equation
cites work: The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
cites work: Introduction to Applied Nonlinear Dynamical Systems and Chaos
cites work: Blow-up of solutions to a degenerate parabolic equation not in divergence form
cites work: Blow-up in a degenerate parabolic equation
DOI: 10.3934/CPAA.2024038
description: scientific article; zbMATH DE number 7896652