Note on asymptotic behavior of weak solutions of the Cauchy problem for some parabolic equations with unbounded coefficients

1979 ◽  
Vol 121 (1) ◽  
pp. 207-215
Author(s):  
Lu-San Chen
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Unbounded Coefficients ◽  
The Cauchy Problem

scholarly journals Properties of Solutions of Parabolic Equations and Inequalities

1961 ◽  
Vol 13 ◽  
pp. 331-345 ◽  
Author(s):  
M. H. Protter
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Adjoint Operator ◽  
Asymptotic Behavior Of Solutions ◽  
The Cauchy Problem ◽  
Image Position ◽  
Parabolic Inequalities

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.We consider parabolic operators in self-adjoint form(1)as well as the non-self-adjoint operator(2)where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.

scholarly journals Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection

2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Mass Conservation ◽  
Degenerate Parabolic Equations ◽  
Conservation Property ◽  
The Cauchy Problem ◽  
Porous Medium Equations ◽  
Degenerate Parabolic

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.

scholarly journals Note on the Uniqueness Proprety of Weak Solutions of Parabolic Equations

1967 ◽  
Vol 29 ◽  
pp. 45-49 ◽  
Author(s):  
Tadashi Kuroda
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Property ◽  
Uniqueness Of Solutions ◽  
Measurable Coefficients ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Aronson proved, in his paper [1], the existence and the uniqueness property of weak solutions of the initial boundary value problem for parabolic equations of second order with measurable coefficients. On the uniqueness of solutions of the Cauchy problem for such equations he also gave some interesting results in [2].

scholarly journals Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection

2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Mass Conservation ◽  
Degenerate Parabolic Equations ◽  
Conservation Property ◽  
The Cauchy Problem ◽  
Porous Medium Equations ◽  
Degenerate Parabolic

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.

scholarly journals The Cauchy Problem of Linear Parabolic Equations with Discontinuous and Unbounded Coefficients

1971 ◽  
Vol 41 ◽  
pp. 33-42 ◽  
Author(s):  
Yoshiaki Ikeda
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Parabolic Equations ◽  
Unbounded Coefficients ◽  
Linear Parabolic Equations ◽  
The Cauchy Problem

In this article we shall prove the uniqueness and existense of a weak solution for the Cauchy problem of linear parabolic equations with discontinuous and unbounded coefficients(1.1)

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