scholarly journals The Cauchy problem for heat equations with exponential nonlinearity

2011 ◽  
Vol 251 (4-5) ◽  
pp. 1172-1194 ◽  
Author(s):  
Norisuke Ioku
Keyword(s):  
Cauchy Problem ◽  
Heat Equations ◽  
Exponential Nonlinearity ◽  
The Cauchy Problem

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.

scholarly journals No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

scholarly journals The Cauchy problem for the heat equation on the plane with a random right part from the Orlicz space

2020 ◽  
pp. 103-109
Author(s):  
A. I. Slyvka-Tylyshchak ◽  
M. M. Mykhasiuk ◽  
O. O. Pohoriliak
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Orlicz Space ◽  
Initial Conditions ◽  
Classical Problem ◽  
Mathematical Physics ◽  
Heat Equations ◽  
Random Initial Conditions ◽  
The Cauchy Problem ◽  
The Right

The heat equation with random conditions is a classical problem of mathematical physics. Recently, a number of works appeared, which in many ways investigated this equation according to the type of random initial conditions. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on the plane with a random right part. We consider the right part as a random function of the Orlicz space. The conditions of existence with probability one classical solution of the problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

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