scholarly journals Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Evolution Equation ◽  
Positive Solutions ◽  
Evolution Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Duality Argument ◽  
Nonlocal Evolution Equation ◽  
Global Nonexistence ◽  
Nonlocal Evolution Equations ◽  
Evolution Systems

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)),x∈RN,whereJ:RN→R+,p>1, and(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

scholarly journals Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Severino Horácio da Silva ◽  
Jocirei Dias Ferreira ◽  
Flank David Morais Bezerra
Keyword(s):  
Evolution Equation ◽  
Lower Semicontinuity ◽  
Evolution Equations ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Functional Parameter ◽  
Normal Hyperbolicity ◽  
Nonlocal Evolution Equations

We show the normal hyperbolicity property for the equilibria of the evolution equation∂m(r,t)/∂t=-m(r,t)+g(βJ*m(r,t)+βh),  h,β≥0,and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameterJ.

Blow-Up of the Solution for some Higher Order Hyperbolic and Neutral Evolution Systems

2011 ◽  
Vol 52-54 ◽  
pp. 121-126
Author(s):  
Ning Chen ◽  
Ji Qian Chen
Keyword(s):  
Evolution Equation ◽  
Mixed Problem ◽  
Finite Time ◽  
Blow Up ◽  
Higher Order ◽  
Neutral Evolution ◽  
Parabolic Evolution Equation ◽  
Evolution Systems

In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher-order nonlinear hyperbolic and parabolic evolution equation in finite time. By introducing the “ blow-up factor ’’, we get some new conclusions, which generalize some results [4]-[5] , [6] .

scholarly journals On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dinh-Ke Tran ◽  
Nhu-Thang Nguyen
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Global Solvability ◽  
Kernel Functions ◽  
Stability Of Solutions ◽  
Nonlinear Perturbations ◽  
Completely Positive ◽  
Positive Kernel ◽  
Nonlocal Evolution Equations

<p style='text-indent:20px;'>We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.</p>

INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS

2010 ◽  
Vol 12 (06) ◽  
pp. 1031-1054 ◽  
Author(s):  
JESÚS GARCÍA-FALSET ◽  
SIMEON REICH
Keyword(s):  
Banach Space ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Sufficient Conditions ◽  
Accretive Operator ◽  
Integral Solution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Evolution Equations ◽  
Integral Solutions

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

scholarly journals On some sufficient conditions for the blow-up solutions of the nonlinear Ginzburg-Landau-Schrödinger evolution equation

2004 ◽  
Vol 2004 (1) ◽  
pp. 23-35 ◽  
Author(s):  
Sh. M. Nasibov
Keyword(s):  
Boundary Condition ◽  
Euclidean Space ◽  
Initial Data ◽  
Evolution Equation ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Homogeneous Boundary Condition ◽  
Dimensional Euclidean Space ◽  
Ginzburg Landau ◽  
The Given

Investigation of the blow-up solutions of the problem in finite time of the first mixed-value problem with a homogeneous boundary condition on a bounded domain ofn-dimensional Euclidean space for a class of nonlinear Ginzburg-Landau-Schrödinger evolution equation is continued. New simple sufficient conditions have been obtained for a wide class of initial data under which collapse happens for the given new values of parameters.

scholarly journals A class of singularly perturbed evolution systems

1994 ◽  
Vol 7 (2) ◽  
pp. 179-190
Author(s):  
N. U. Ahmed
Keyword(s):  
Continuous Dependence ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Singularly Perturbed ◽  
Evolution Operators ◽  
Semilinear Systems ◽  
Existence Of Periodic Solutions ◽  
Semigroup Generators ◽  
Evolution Systems

In this paper we study a class of evolution equations where the semigroup generators are singularly perturbed by a nonnegative real valued function of time. Sufficient conditions for existence of evolution operators and their compactness are given including continuous dependence on the perturbation. Further, for a coupled system of singularly perturbed semilinear systems in two Banach spaces, existence of periodic solutions and their stability are studied.

scholarly journals Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations

2020 ◽  
Vol 9 (1) ◽  
pp. 1569-1591
Author(s):  
Menglan Liao ◽  
Qiang Liu ◽  
Hailong Ye
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Finite Time ◽  
Evolution Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Potential Well

Abstract In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

Sign in / Sign up

Export Citation Format

Share Document