scholarly journals A non-linear hyperbolic equation

1980 ◽  
Vol 3 (3) ◽  
pp. 505-520 ◽  
Author(s):  
Eliana Henriques de Brito
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Real Number ◽  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Linear Hyperbolic Equation ◽  
Non Linear

In this paper the following Cauchy problem, in a Hilbert spaceH, is considered:(I+λA)u″+A2u+[α+M(|A12u|2)]Au=fu(0)=u0u′(0)=u1Mandfare given functions,Aan operator inH, satisfying convenient hypothesis,λ≥0andαis a real number.Foru0in the domain ofAandu1in the domain ofA12, ifλ>0, andu1inH, whenλ=0, a theorem of existence and uniqueness of weak solution is proved.

Cauchy Problem for a Quasi-Linear Hyperbolic Equation with Closed Support of Data

2013 ◽  
Vol 193 (3) ◽  
pp. 364-368 ◽  
Author(s):  
G. Baghaturia ◽  
J. Gvazava ◽  
M. Menteshashvili
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Linear Hyperbolic Equation

scholarly journals Existence, Uniqueness and C −Differentiability of Solutions in a Non-linear Model of Cancerous Tumor

2018 ◽  
Vol 10 (6) ◽  
pp. 43
Author(s):  
Gossan D. Pascal Gershom ◽  
Yoro Gozo ◽  
Bailly Balé
Keyword(s):  
Mathematical Modeling ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Linear Operators ◽  
System Of Nonlinear Equations ◽  
Small Perturbations ◽  
Direct Mapping ◽  
Non Linear ◽  
Cancer Tumor ◽  
Differentiability Of Solutions

In this paper, we prove the existence and uniqueness of the weak solution of a system of nonlinear equations involved in the mathematical modeling of cancer tumor growth with a non homogeneous divergence condition. We also present  a new concept of generalized differentiation of non linear operators : C-differentiability. Through this notion, we also prove the uniqueness and the C-differentiability of the solution when the system is perturbed by a certain number of parameters. Two results have been established. In the first one, differentiability is according to Fréchet. The proof is given uses the theorem of reciprocal functions in Banach spaces. First of all, we give the proof of strict differentiability of a direct mapping, according to Fréchet. In the second result, differentiability is understood in a weaker sense than that of Fréchet. For the proof we use Hadamard's theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

Oscillation of second-order hyperbolic equations with non-integrable coefficients

1982 ◽  
Vol 91 (3-4) ◽  
pp. 305-313 ◽  
Author(s):  
Wu-Teh Hsiang ◽  
Man Kam Kwong
Keyword(s):  
Hyperbolic Equation ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Case ◽  
Linear Hyperbolic Equation ◽  
Initial Values ◽  
Non Linear ◽  
Image Position

SynopsisSome sufficient conditions are obtained on the coefficient g and the initial values Φ and ψfor the solution ot the non-linear hyperbolic equationto change sign in the first quadrant. An example is given to show that is not sufficient in the linear case.

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