scholarly journals The Global Existence of Nonlinear Evolutionary Equation with Small Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Xunwu Yin
Keyword(s):  
Global Existence ◽  
Local Existence ◽  
Evolutionary Equation ◽  
Fractional Powers ◽  
Gronwall’S Inequality ◽  
Sectorial Operators ◽  
Local Existence And Uniqueness ◽  
Small Delay ◽  
Fixed Point Principle ◽  
Nonlinear Evolutionary Equation

We investigate the global existence of the delayed nonlinear evolutionary equation∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers spaceXα. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.

Global existence of solutions for a nonlinear size-structured population model with distributed delay in the recruitment

2015 ◽  
Vol 26 (10) ◽  
pp. 1550085 ◽  
Author(s):  
Meng Bai ◽  
Shihe Xu
Keyword(s):  
Global Existence ◽  
Population Model ◽  
Initial Conditions ◽  
Local Existence ◽  
Extension Theorem ◽  
Distributed Delay ◽  
Structured Population Model ◽  
Structured Population ◽  
Global Existence Of Solutions ◽  
Local Existence And Uniqueness

In this paper, we study a nonlinear size-structured population model with distributed delay in the recruitment. The delayed problem is reduced into an abstract initial value problem of an ordinary differential equation in a Banach space by using the semigroup techniques. The local existence and uniqueness of solution as well as the continuous dependence on initial conditions are obtained by using the general theory of quasi-linear evolution equations in Banach spaces, while the global existence of solution is obtained by the estimates of the solution and the extension theorem.

scholarly journals On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Christian Rohde ◽  
Hao Tang
Keyword(s):  
Global Existence ◽  
Wave Breaking ◽  
Evolution Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
Nonlinear Noise ◽  
Pathwise Solutions

AbstractWe consider a class of stochastic evolution equations that include in particular the stochastic Camassa–Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $$H^s$$ H s with $$s>3/2$$ s > 3 / 2 . Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.

scholarly journals The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

2019 ◽  
Vol 26 (6) ◽  
Author(s):  
Diego Alonso-Orán ◽  
Aythami Bethencourt de León ◽  
So Takao
Keyword(s):  
Global Existence ◽  
Existence And Uniqueness ◽  
Stochastic Equation ◽  
Burgers Equation ◽  
Blow Up ◽  
Local Existence ◽  
Smooth Solutions ◽  
Local Existence And Uniqueness ◽  
Stochastic Transport ◽  
Global Existence And Uniqueness

Abstract In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

Local existence and blow-up criterion for the Boussinesq equations

1997 ◽  
Vol 127 (5) ◽  
pp. 935-946 ◽  
Author(s):  
Dongho Chae ◽  
Hee-Seok Nam
Keyword(s):  
Global Existence ◽  
External Force ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Passive Scalar ◽  
Smooth Solutions ◽  
Local Existence And Uniqueness ◽  
Blow Up Criterion

SynopsisIn this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

scholarly journals On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

2020 ◽  
Vol 30 (4) ◽  
pp. 572-584
Author(s):  
D.K. Durdiev ◽  
J.Z. Nuriddinov
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Variable Coefficient ◽  
Direct Problem ◽  
Local Existence ◽  
Problem Solution ◽  
Volterra Type ◽  
Integral Term ◽  
Local Existence And Uniqueness ◽  
Coefficient Of Thermal Conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

scholarly journals Local and Global Existence of Solution for Love Type Waves with Past History

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1998
Author(s):  
Mohamed Biomy ◽  
Khaled Zennir ◽  
Ahmed Himadan
Keyword(s):  
Existence And Uniqueness ◽  
Past History ◽  
Local Existence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linearization Method ◽  
Weak Compactness ◽  
Existence Of Solution ◽  
Local Existence And Uniqueness ◽  
Initial Boundary Value

In this paper, we consider an initial boundary value problem for nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.

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