scholarly journals The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior

2016 ◽  
Vol 443 (1) ◽  
pp. 265-294 ◽  
Author(s):  
Claudianor O. Alves ◽  
Sergey Shmarev ◽  
Jacson Simsen ◽  
Mariza S. Simsen
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Sobolev Spaces ◽  
Parabolic Equations ◽  
The Cauchy Problem ◽  
Weighted Variable

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

2018 ◽  
Vol 111 (1) ◽  
pp. 43-68
Author(s):  
Sergey Shmarev ◽  
Jacson Simsen ◽  
Mariza Stefanello Simsen ◽  
Marcos Roberto T. Primo
Keyword(s):  
Asymptotic Behavior ◽  
Sobolev Spaces ◽  
Parabolic Equations ◽  
Weighted Variable

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 Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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