A note on the Cauchy problem to a class of nonlinear dispersive equations with singular initial data

2000 ◽  
Vol 42 (2) ◽  
pp. 251-270 ◽  
Author(s):  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Dispersive Equations ◽  
Nonlinear Dispersive Equations ◽  
The Cauchy Problem ◽  
Singular Initial Data

scholarly journals The Cauchy Problem for a Fifth-Order Dispersive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjun Wang ◽  
Yongqi Liu ◽  
Yongqiang Chen
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Order Equation ◽  
Dispersive Equations ◽  
Dispersive Equation ◽  
The Cauchy Problem ◽  
Fifth Order ◽  
Well Posed

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev spaceHs(R)withs≥1/4. We also establish the ill-posedness for the initial data inHs(R)withs<1/4. Thus, the regularity requirement for the fifth-order dispersive equationss≥1/4is sharp.

HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA

2006 ◽  
Vol 08 (03) ◽  
pp. 331-354 ◽  
Author(s):  
V. A. GALAKTIONOV ◽  
A. E. SHISHKOV
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Quasilinear Parabolic Equation ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Semilinear Heat Equation ◽  
Basic Model ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Quasilinear Parabolic

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,p is the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN × R+ with arbitrary singular initial data u0 ≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

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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