scholarly journals Well-Posedness Results for the Continuum Spectrum Pulse Equation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1006 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Optical Waveguides ◽  
Classical Solutions ◽  
Well Posedness ◽  
Third Order ◽  
Electrical Field ◽  
Continuum Spectrum ◽  
Linearly Polarized ◽  
The Cauchy Problem ◽  
Nonlocal Nonlinear ◽  
The Continuum

The continuum spectrum pulse equation is a third order nonlocal nonlinear evolutive equation related to the dynamics of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides. In this paper, the well-posedness of the classical solutions to the Cauchy problem associated with this equation is proven.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 170 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Phase Transitions ◽  
High Order ◽  
Classical Solutions ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Oil Water ◽  
Surfactant Systems ◽  
Dynamics Of Phase Transitions

High order convective Cahn-Hilliard type equations describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

The well-posedness theory for Euler–Poisson fluids with non-zero heat conduction

2014 ◽  
Vol 11 (04) ◽  
pp. 679-703
Author(s):  
Jiang Xu
Keyword(s):  
Heat Conduction ◽  
Besov Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Classical Solutions ◽  
Well Posedness ◽  
Poisson Equations ◽  
The Cauchy Problem ◽  
Critical Besov Spaces ◽  
Temperature Equation

This paper is devoted to the Euler–Poisson equations for fluids with non-zero heat conduction, arising in semiconductor science. Due to the thermal effect of the temperature equation, the local well-posedness theory by Xu and Kawashima (2014) for generally symmetric hyperbolic systems in spatially critical Besov spaces does not directly apply. To deal with this difficulty, we develop a generalized version of the Moser-type inequality by using Bony's decomposition. With a standard iteration argument, we then establish the local well-posedness of classical solutions to the Cauchy problem for intial data in spatially Besov spaces.

scholarly journals Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Classical Solutions ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation ◽  
Sivashinsky Equation

AbstractThe Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

scholarly journals Well-posedness result for the Kuramoto–Velarde equation

2021 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Cauchy Problem ◽  
Initial Data ◽  
Diffusion Reaction ◽  
Microgravity Environment ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Fronts ◽  
The Cauchy Problem ◽  
Time Variations

AbstractThe Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

scholarly journals General decay and well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation with memory

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1745-1773
Author(s):  
Salah Boulaaras ◽  
Abdelbaki Choucha ◽  
Djamel Ouchenane
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Generalized Solution ◽  
General Decay ◽  
Small Data ◽  
Well Posedness ◽  
Third Order ◽  
The Cauchy Problem ◽  
With Memory

In this paper, we consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan-Moore-Gibson-Thompson (JMGT) equation with the presence of both memory. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result, and we show a local existence result in appropriate function spaces. Finally, we prove a global existence result for small data, and we prove the uniqueness of the generalized solution.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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