Sufficient conditions for stabilization of solutions of the cauchy problem for nondivergent parabolic equations with lower-order coefficients

2010 ◽  
Vol 171 (1) ◽  
pp. 46-57 ◽  
Author(s):  
Vasiliĭ Nikolaevich Denisov
Keyword(s):  
Cauchy Problem ◽  
Lower Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
The Cauchy Problem

scholarly journals On Large-Time Behavior of Solutions of Parabolic Equations

2020 ◽  
Vol 66 (1) ◽  
pp. 1-155
Author(s):  
Vasiliy N. Denisov
Keyword(s):  
Growth Rate ◽  
Cauchy Problem ◽  
Boundary Value Problem ◽  
Lower Order ◽  
Parabolic Equations ◽  
Initial Function ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Time Behavior ◽  
The Cauchy Problem

We study the stabilization of solutions of the Cauchy problem for second-order parabolic equations depending on the behavior of the lower-order coefficients of equations at the infinity and on the growth rate of initial functions. We also consider the stabilization of solution of the first boundary-value problem for a parabolic equation without lower-order coefficients depending on the domain Q where the initial function is defined for t =0. In the first chapter, we study sufficient conditions for uniform in x on a compact K RN stabilization to zero of the solution of the Cauchy problem with divergent elliptic operator and coefficients independent of t and depending only on x. We consider classes of initial functions: bounded in RN, with power growth rate at the infinity in RN, with exponential order at the infinity. Using examples, we show that sufficient conditions are sharp and, moreover, do not allow the uniform in RN stabilization to zero of the solution of the Cauchy problem. In the second chapter, we study the Cauchy problem with elliptic nondivergent operator and coefficients depending on x and t. In different classes of growing initial functions we obtain exact sufficient conditions for stabilization of solutions of the corresponding Cauchy problem uniformly in x on any compact K in RN. We consider examples proving the sharpness of these conditions. In the third chapter, for the solution of the first boundary-value problem without lower-order terms, we obtain necessary and sufficient conditions of uniform in x on any compact in Q stabilization to zero in terms of the domain RN \ Q where Q is the definitional domain of the initial function for t =0. We establish the power estimate for the rate of stabilization of the solution of the boundary-value problem with bounded initial function in the case where RN \ Q is a cone for t =0.

scholarly journals On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Term

2017 ◽  
Vol 63 (4) ◽  
pp. 586-598
Author(s):  
V N Denisov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Half Space ◽  
Lower Order ◽  
Parabolic Equations ◽  
Initial Function ◽  
Order Term ◽  
Sufficient Conditions ◽  
Lower Order Term ◽  
The Cauchy Problem

In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).

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.

On stabilisation of solutions of the Cauchy problem for parabolic equations

1976 ◽  
Vol 76 (1) ◽  
pp. 43-53 ◽  
Author(s):  
S. Kamin (Kamenomostskaya)
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Cauchy Problem ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThe author considers the solution of the Cauchy problem for an equationgiving necessary and sufficient conditions for the existence of

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

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