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.

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 of Solutions for a System of Nonlinear Viscoelastic Equations with Damping Terms in Rn

2009 ◽  
Vol 64 (3-4) ◽  
pp. 180-184
Author(s):  
Wenjun Liu ◽  
Shengqi Yub
Keyword(s):  
Initial Data ◽  
Differential Inequality ◽  
Blow Up ◽  
Coupled System ◽  
Initial Energy ◽  
Relaxation Functions ◽  
Nonlinear Viscoelastic ◽  
Linear Damping ◽  
Viscoelastic Equations ◽  
Damping And Source Terms

Abstract We consider a coupled system of nonlinear viscoelastic equations with linear damping and source terms. Under suitable conditions of the initial data and the relaxation functions, we prove a finitetime blow-up result with vanishing initial energy by using the modified energy method and a crucial lemma on differential inequality

scholarly journals A general blow-up result for a degenerate hyperbolic inequality in an exterior domain

2021 ◽  
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Boundary Conditions ◽  
Critical Exponent ◽  
Exterior Domain ◽  
Critical Case ◽  
Blow Up ◽  
Unified Approach ◽  
Dirichlet Type ◽  
Open Questions ◽  
Neumann Type ◽  
Type Boundary

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

scholarly journals Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

2008 ◽  
Vol 244 (11) ◽  
pp. 2693-2740 ◽  
Author(s):  
J. Emile Rakotoson ◽  
J. Michel Rakotoson ◽  
Cédric Verbeke
Keyword(s):  
Thin Films ◽  
Initial Data ◽  
Global Existence ◽  
Blow Up ◽  
Higher Order

scholarly journals On Removable Singularities of Solutions of Higher-Order Differential Inequalities

2020 ◽  
Vol 20 (2) ◽  
pp. 385-397
Author(s):  
A. A. Kon’kov ◽  
A. E. Shishkov
Keyword(s):  
Unit Ball ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Removable Singularity ◽  
Higher Order ◽  
Differential Inequalities ◽  
Removable Singularities

AbstractWe obtain sufficient conditions for solutions of the mth-order differential inequality\sum_{|\alpha|=m}\partial^{\alpha}a_{\alpha}(x,u)\geq f(x)g(|u|)\quad\text{in % }B_{1}\setminus\{0\}to have a removable singularity at zero, where {a_{\alpha}}, f, and g are some functions, and {B_{1}=\{x:|x|<1\}} is a unit ball in {{\mathbb{R}}^{n}}. We show in some examples the sharpness of these conditions.

scholarly journals Blow-up for the wave equation with nonlinear source and boundary damping terms

2015 ◽  
Vol 145 (4) ◽  
pp. 759-778 ◽  
Author(s):  
Alessio Fiscella ◽  
Enzo Vitillaro
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Initial Data ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Nonlinear Dissipation ◽  
Typical Problem ◽  
Nonlinear Source ◽  
Boundary Damping

The paper deals with blow-up for the solutions of an evolution problem consisting in a semilinear wave equation posed in a boundedC1,1open subset of ℝn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied iswhere∂Ω=Γ0∪Γ1,Γ0∩Γ1= ∅,σ(Γ0) > 0, 2 <p≤ 2(n− 1)/(n− 2) (whenn≥ 3),m> 1,α∈L∞(Γ1),α≥ 0 andβ≥ 0. The initial data are posed in the energy space.The aim of the paper is to improve previous blow-up results concerning the problem.

Global Sign-changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Victor A. Galaktionov ◽  
Enzo Mitidieri ◽  
Stanislav I. Pohozaev
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Initial Function ◽  
Blow Up ◽  
Higher Order ◽  
Semilinear Heat Equation ◽  
Sign Changing Solutions

AbstractA detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita rangewith bounded integrable initial data u(x, 0) = uwith the same initial data u∫ ui.e., as for (0.1), any such arbitrarily small initial function u

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