The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations

2012 ◽  
Vol 55 (3) ◽  
pp. 623-631
Author(s):  
Jiaqing Pan
Keyword(s):  
Cauchy Problem ◽  
Compact Subset ◽  
Continuous Dependence ◽  
Diffusion Equations ◽  
Fast Diffusion ◽  
The Cauchy Problem ◽  
Image Position ◽  
Fast Diffusion Equations

AbstractIn this paper, we consider the Cauchy problemWe will prove that(i) for mc < m,m0 < 1, |u(x, t, m)–u(x, t, m0)| → 0 as m → m0 uniformly on every compact subset of ℝN × ℝ+, where ;(ii) there is a C* that explicitly depends on m such that

Existence and uniqueness of the global admissible solution for a viscoelastic model with relaxation

1996 ◽  
Vol 126 (5) ◽  
pp. 1113-1132 ◽  
Author(s):  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Viscoelastic Model ◽  
Admissible Solution ◽  
Existence Problem ◽  
Large Initial Data ◽  
The Cauchy Problem ◽  
Definition Of ◽  
Image Position

This paper examines the Cauchy problem for a viscoelastic model with relaxationwith discontinuous, large initial data, where ½ ≦ μ <1, δ > 0 are constants. We first give a definition of admissible (or entropic) solutions to the system. Under this definition, we prove the existence, uniqueness and continuous dependence of the global admissible solution for the system. Our methods are essentially due to Kruzkov, and the requirement that f(u) is not badly degenerate (more precisely, meas {x: f″(x) = 0} = 0), needed previously when considering the global existence problem for the same system, is removed.

Poincaré inequalities for linearizations of very fast diffusion equations

Nonlinearity ◽  
2002 ◽  
Vol 15 (3) ◽  
pp. 565-580 ◽  
Author(s):  
J A Carrillo ◽  
C Lederman ◽  
P A Markowich ◽  
G Toscani
Keyword(s):  
Diffusion Equations ◽  
Fast Diffusion ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Fast Diffusion Equations

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

Extinction behaviour for fast diffusion equations with absorption

2001 ◽  
Vol 43 (8) ◽  
pp. 943-985 ◽  
Author(s):  
Raúl Ferreira ◽  
Juan Luis Vazquez
Keyword(s):  
Diffusion Equations ◽  
Fast Diffusion ◽  
Fast Diffusion Equations

scholarly journals Explicit solutions for a system of coupled Lyapunov differential matrix equations

1987 ◽  
Vol 30 (3) ◽  
pp. 427-434 ◽  
Author(s):  
L. Jodar ◽  
M. Mariton
Keyword(s):  
Cauchy Problem ◽  
Explicit Expression ◽  
Linear Quadratic ◽  
The Cauchy Problem ◽  
Matrix Differential Equations ◽  
Lyapunov Matrix ◽  
Error Accumulation ◽  
Matrix Differential ◽  
Image Position ◽  
Differential Matrix

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

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