Existence and Uniqueness of Global Solutions of Regular Characteristic Cauchy Problems

1993 ◽  
Vol 161 (1) ◽  
pp. 75-93
Author(s):  
Michael Reissig
Keyword(s):  
Existence And Uniqueness ◽  
Global Solutions ◽  
Cauchy Problems

scholarly journals Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I

1977 ◽  
Vol 16 (3) ◽  
pp. 379-414 ◽  
Author(s):  
Aleksander Glikson
Keyword(s):  
Boltzmann Equation ◽  
Existence And Uniqueness ◽  
Broad Class ◽  
Physical Space ◽  
Uniqueness Result ◽  
Cauchy Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Boltzmann Gas ◽  
Definition Of

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

scholarly journals PullbackD-Attractor of Coupled Rod Equations with Nonlinear Moving Heat Source

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Danxia Wang ◽  
Jianwen Zhang ◽  
Yinzhu Wang
Keyword(s):  
Heat Source ◽  
Boundary Conditions ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Initial Conditions ◽  
Global Solutions ◽  
Absorbing Set ◽  
Moving Heat Source ◽  
Condition C

We consider the pullbackD-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullbackD-absorbing set is obtained. By proving the properties of compactness about the nonlinear operatorg1(·),g2(·), and then proving the pullbackD-condition (C), the existence of the pullbackD-attractor of the equations previously mentioned is given.

scholarly journals Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Daewook Kim ◽  
Dojin Kim ◽  
Keum-Shik Hong ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Existence And Uniqueness ◽  
Global Solutions ◽  
Growth Conditions ◽  
Decay Rates ◽  
Decay Estimates ◽  
Nonlinear Dissipation ◽  
Kirchhoff Type ◽  
Type Wave ◽  
Energy Decay Rates

The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form under suitable assumptions on . Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipationg. Lastly, numerical simulations in order to verify the analytical results are given.

scholarly journals Global existence for the generalised 2D Ginzburg-Landau equation

2003 ◽  
Vol 44 (3) ◽  
pp. 381-392 ◽  
Author(s):  
Hongjun Gao ◽  
Keng-Huat Kwek
Keyword(s):  
Existence And Uniqueness ◽  
Landau Equation ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Ginzburg Landau ◽  
Spatial Variables ◽  
Type Complex ◽  
Ginzburg Landau Equation ◽  
Fifth Order ◽  
Formation Systems

AbstractGinzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.

scholarly journals On The Solvability of an Inverse Fractional Abstract Cauchy Problems with Conformable Derivative

2021 ◽  
Author(s):  
Lahcen Rabhi ◽  
Mohammed AL HORANI ◽  
R. Khalil
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Parabolic Partial Differential Equations ◽  
Cauchy Problems ◽  
Conformable Derivative ◽  
Equivalent Statement ◽  
Special Cases ◽  
Abstract Cauchy Problems

In this paper, we discuss the solvability of fractional inverse problem for the conformable derivative in Banach space. We establish an equivalent statement of the existence and uniqueness of solution using fractional semigroup. Some special cases of the inverse problem are studied. An application is given to study an inverse problem in a suitable Sobolev space for fractional parabolic partial differential equations with unknown source functions.

scholarly journals Boundary linear stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Nejib Smaoui ◽  
Boumediène Chentouf ◽  
Ala’ Alalabi
Keyword(s):  
Exponential Stability ◽  
Numerical Simulations ◽  
Boundary Control ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Global Solutions ◽  
Spatial Variable ◽  
Stabilization Problem ◽  
De Vries ◽  
Adaptive Boundary Control

Abstract The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in $[0,1]$ [ 0 , 1 ] . First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in $L^{2} (0,1)$ L 2 ( 0 , 1 ) . Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.

scholarly journals The asymptotic behaviour of fractional lattice systems with variable delay

2019 ◽  
Vol 22 (3) ◽  
pp. 681-698
Author(s):  
Linfang Liu ◽  
Tomás Caraballo ◽  
Peter E. Kloeden
Keyword(s):  
Asymptotic Behaviour ◽  
Existence And Uniqueness ◽  
Time Delays ◽  
Lipschitz Constant ◽  
Time Derivative ◽  
Global Solutions ◽  
Variable Delay ◽  
Lattice Systems ◽  
Functional Differential ◽  
Fractional Functional

Abstract The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a certain condition on the Lipschitz constant.

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