scholarly journals On a Class of Multitime Evolution Equations with Nonlocal Initial Conditions

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
F. Zouyed ◽  
F. Rebbani ◽  
N. Boussetila
Keyword(s):  
Evolution Equation ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
A Priori Estimates ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlocal Initial Conditions ◽  
Priori Estimates

The existence and uniqueness of the strong solution for a multitime evolution equation with nonlocal initial conditions are proved. The proof is essentially based on a priori estimates and on the density of the range of the operator generated by the considered problem.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

scholarly journals The Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition

2020 ◽  
Vol 11 (4) ◽  
pp. 1991-2022
Author(s):  
Annamaria Barbagallo ◽  
Vincenzo Esposito
Keyword(s):  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Existence And Uniqueness Results ◽  
Mixed Problems ◽  
Uniqueness Results ◽  
Hyperbolic Operators ◽  
Priori Estimates

Abstract The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.

scholarly journals A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 852 ◽  
Author(s):  
Jieqiong Shen ◽  
Bin Li
Keyword(s):  
Nonlinear System ◽  
Strong Solution ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Upper Bounds ◽  
Biological Transport ◽  
Transport Networks ◽  
Cross Diffusion ◽  
Priori Estimates

In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.

A THEOREM DUAL TO YAMADA–WATANABE THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 10 (03) ◽  
pp. 367-374 ◽  
Author(s):  
HUIJIE QIAO
Keyword(s):  
Evolution Equation ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Variational Approach ◽  
Evolution Equations ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniqueness In Law ◽  
Strong Existence

In this paper, we prove that uniqueness in law and strong existence for a stochastic evolution equation [Formula: see text] imply existence and uniqueness of a strong solution in the framework of the variational approach. This result seems to be dual to Yamada–Watanabe theorem in [7].

A difference scheme for equations of ocean dynamics on unstructured grids

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexey V. Drutsa
Keyword(s):  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Large Scale ◽  
A Priori Estimates ◽  
Unstructured Grids ◽  
A Priori ◽  
Ocean Dynamics ◽  
Priori Estimates ◽  
Uniqueness Of The Solution ◽  
Grid Operators

AbstractA difference scheme on unstructured grids is constructed for the system of equations of large scale ocean dynamics. The properties of the grid problem and grid operators are studied, in particular, a series of a priori estimates and the theorem on existence and uniqueness of the solution are proved.

A PRIORI BOUNDS ON SPATIAL MOTIONS OF INCOMPRESSIBLE NONLINEARLY ELASTIC RODS

2006 ◽  
Vol 03 (03) ◽  
pp. 481-504 ◽  
Author(s):  
STUART S. ANTMAN
Keyword(s):  
Evolution Equations ◽  
A Priori Estimates ◽  
Hyperbolic Equations ◽  
A Priori ◽  
Careful Analysis ◽  
Elastic Rods ◽  
Governing Equations ◽  
Mathematical Structures ◽  
Priori Estimates ◽  
Nonlinearly Elastic

The geometrically exact quasilinear evolution equations governing the spatial motion of incompressible rods and their specializations to the hyperbolic equations governing nonlinearly elastic rods have novel mathematical structures strikingly different from those for the equations governing the motion of compressible rods. The main objectives of this paper are to formulate the governing equations, an exercise requiring the solution of a sequence of semilinear hyperbolic equations of first order, and to derive a priori estimates for certain strain variables, ensuring that they cannot reach geometrically prohibited ranges in finite time. This process exhibits a new system of physically important quasilinear equations worthy of careful analysis.

On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type

1973 ◽  
Vol 16 (1) ◽  
pp. 137-141
Author(s):  
K. A. Zischka
Keyword(s):  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Equations ◽  
Priori Estimates ◽  
The Given ◽  
Uniqueness Of A Solution

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

ASYMPTOTIC BEHAVIOR OF STOCHASTIC DISSIPATIVE QUANTUM ZAKHAROV EQUATIONS

2013 ◽  
Vol 13 (02) ◽  
pp. 1250016 ◽  
Author(s):  
YANFENG GUO ◽  
BOLING GUO ◽  
DONGLONG LI
Keyword(s):  
Asymptotic Behavior ◽  
White Noise ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Galerkin Approximation ◽  
Uniqueness Of Solutions ◽  
Asymptotic Behaviors ◽  
Priori Estimates

The stochastic dissipative quantum Zakharov equations with white noise are studied. The existence and uniqueness of solutions are obtained by using the standard Galerkin approximation method on the basis of the time uniform a priori estimates in various spaces. Moreover, the asymptotic behaviors of the solutions for the stochastic dissipative quantum Zakharov equations with white noise are also investigated.

scholarly journals Two-step Bdf Time Discretisation of Nonlinear Evolution Problems Governed by Monotone Operators with Strongly Continuous Perturbations

2009 ◽  
Vol 9 (1) ◽  
pp. 37-62 ◽  
Author(s):  
E. Emmrich
Keyword(s):  
Evolution Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Monotone Operators ◽  
Continuous Operator ◽  
Time Dependent ◽  
Uniform Grid ◽  
Differentiation Formula ◽  
Time Discretisation ◽  
Priori Estimates

Abstract The time discretisation of the initial-value problem for a first-order evolution equation by the two-step backward differentiation formula (BDF) on a uniform grid is analysed. The evolution equation is governed by a time-dependent monotone operator that might be perturbed by a time-dependent strongly continuous operator. Well-posedness of the numerical scheme, a priori estimates, convergence of a piecewise polynomial prolongation, stability as well as smooth-data error estimates are provided relying essentially on an algebraic relation that implies the G-stability of the two-step BDF with constant time steps.

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