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

2021 ◽  
Author(s):  
V. N. Denisov
Keyword(s):  
Cauchy Problem ◽  
Lower Order ◽  
Parabolic Equations ◽  
The Cauchy Problem ◽  
Lower Order Terms

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.

Existence results for a class of nonlinear parabolic equations with two lower order terms

2014 ◽  
Vol 41 (2) ◽  
pp. 207-219
Author(s):  
Ahmed Aberqi ◽  
Jaouad Bennouna ◽  
M. Hammoumi ◽  
Mounir Mekkour ◽  
Ahmed Youssfi
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Existence Results ◽  
Nonlinear Parabolic ◽  
Lower Order Terms
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