scholarly journals Well-Posedness of the Cauchy Problem for Hyperbolic Equations with Non-Lipschitz Coefficients

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Akbar B. Aliev ◽  
Gulnara D. Shukurova
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equations ◽  
Well Posedness ◽  
Singular Coefficients ◽  
The Cauchy Problem ◽  
Finite Smoothness

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients. We prove well-posedness of the corresponding Cauchy problem in some functional spaces. These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothness with respect to variables corresponding to singular coefficients.

HOMOGENEOUS HYPERBOLIC EQUATIONS WITH COEFFICIENTS DEPENDING ON ONE SPACE VARIABLE

2007 ◽  
Vol 04 (03) ◽  
pp. 533-553 ◽  
Author(s):  
SERGIO SPAGNOLO ◽  
GIOVANNI TAGLIALATELA
Keyword(s):  
Cauchy Problem ◽  
Space Variable ◽  
Hyperbolic Equations ◽  
Diagonal Matrix ◽  
Well Posedness ◽  
Characteristic Roots ◽  
The Cauchy Problem

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition [Formula: see text] we succeed in constructing a smooth symmetrizer which behaves like a diagonal matrix: this allows us to prove the well-posedness in [Formula: see text].

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.

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.

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