scholarly journals New Fujita type results for quasilinear parabolic differential inequalities with gradient dissipation terms

2021 ◽  
Vol 6 (10) ◽  
pp. 11482-11493
Author(s):  
Xiaomin Wang ◽  
◽  
Zhong Bo Fang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Critical Exponents ◽  
Differential Inequality ◽  
Source Term ◽  
Differential Inequalities ◽  
Dissipation Term ◽  
Slow Decay ◽  
Nonexistence Of Solutions ◽  
Quasilinear Parabolic

<abstract><p>This paper deals with the new Fujita type results for Cauchy problem of a quasilinear parabolic differential inequality with both a source term and a gradient dissipation term. Specially, nonnegative weights may be singular or degenerate. Under the assumption of slow decay on initial data, we prove the existence of second critical exponents $ \mu^{*} $, such that the nonexistence of solutions for the inequality occurs when $ \mu &lt; \mu^{*} $.</p></abstract>

Blow-Up Results for Higher-Order Evolution Differential Inequalities in Exterior Domains

2019 ◽  
Vol 19 (2) ◽  
pp. 375-390
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Initial Data ◽  
Critical Exponents ◽  
Differential Inequality ◽  
Exterior Domain ◽  
Blow Up ◽  
Neumann Boundary ◽  
Higher Order ◽  
Differential Inequalities ◽  
Exterior Domains ◽  
Unified Approach

AbstractWe consider a higher-order evolution differential inequality in an exterior domain of {\mathbb{R}^{N}}, {N\geq 3}, with Dirichlet and Neumann boundary conditions. Using a unified approach, we obtain the critical exponents in the sense of Fujita for the considered problems. Moreover, the behavior of the solutions with respect to the initial data is discussed.

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 Functional Differential Inequalities with Unbounded Delay

2005 ◽  
Vol 12 (2) ◽  
pp. 237-254
Author(s):  
Zdzisław Kamont ◽  
Adam Nadolski
Keyword(s):  
Differential Inequality ◽  
Continuous Dependence ◽  
Functional Differential Equations ◽  
Uniqueness Result ◽  
Classical Solutions ◽  
Differential Inequalities ◽  
Unbounded Delay ◽  
Several Variables ◽  
Functional Differential ◽  
Continuous Dependence Of Solutions

Abstract We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perron-type uniqueness result and a result on continuous dependence of solutions on given functions for partial functional differential equations with unbounded delay. We consider classical solutions on the Haar pyramid.

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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