ON SOME CHARACTERISTIC CAUCHY PROBLEMS

2021 ◽  
Vol 10 (8) ◽  
pp. 3055-3062
Author(s):  
Jean-André Marti
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Existence And Uniqueness ◽  
Characteristic Curve ◽  
Cauchy Problems ◽  
Uniqueness Of The Solution ◽  
Ill Posed

By means of some regularizations for an ill posed Cauchy problem, we define an associated generalized problem and discuss the conditions for solvability of it. To illustrate this, starting from the semilinear unidirectional wave equation with data given on a characteristic curve, we show existence and uniqueness of the solution in convenient generalized algebras.

scholarly journals ON STOCHASTIC GENERALIZED FUNCTIONS

2011 ◽  
Vol 14 (02) ◽  
pp. 237-260 ◽  
Author(s):  
PEDRO CATUOGNO ◽  
CHRISTIAN OLIVERA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Generalized Functions ◽  
Uniqueness Of The Solution ◽  
Stochastic Cauchy Problem

In this work we introduce a new algebra of stochastic generalized functions. The regular Hida distributions in [Formula: see text] are embedded in this algebra via their chaos expansions. As an application, we prove the existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

scholarly journals Existence, blow-up and exponential decay estimates for the nonlinear Kirchhoff-Carrier wave equation in an annular with nonhomogeneous Dirichlet conditions

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5561-5588 ◽  
Author(s):  
le Son ◽  
Le Ngoc ◽  
Nguyen Long
Keyword(s):  
Wave Equation ◽  
Exponential Decay ◽  
Existence And Uniqueness ◽  
Lyapunov Functional ◽  
Blow Up ◽  
Initial Energy ◽  
Decay Estimates ◽  
Carrier Wave ◽  
Dirichlet Conditions ◽  
Uniqueness Of The Solution

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

scholarly journals DIRCHLET PROBLEM FOR A NONLOCAL WAVE EQUATION WITH RIEMANN-LIOUVILLE DERIVATIVE

2019 ◽  
pp. 6-11
Author(s):  
О.Х. Масаева
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Liouville Derivative ◽  
Uniqueness Of The Solution

Доказано существование и единственность решения задачи Дирихле для уравнения второго порядка с дробной производной. Исследуемое уравнение переходит в волновое уравнение при целом значении порядка дробной производной The existence and uniqueness of the solution to Dirichlet problem for a secondorder equation with a fractional derivative is proved. The equation under study is a wave equation for a integer value of the order of the fractional derivative.

scholarly journals Global classical solutions to the Cauchy problem for a nonlinear wave equation

1998 ◽  
Vol 21 (3) ◽  
pp. 533-548 ◽  
Author(s):  
Haroldo R. Clark
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Classical Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Global Classical Solutions ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem{u″+M(|A12u|2)Au=0   in   ]0,T[u(0)=u0,       u′(0)=u1,whereu′is the derivative in the sense of distributions and|A12u|is theH-norm ofA12u. We prove the existence and uniqueness of global classical solution.

scholarly journals Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations

1996 ◽  
Vol 19 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Pierluigi Colli ◽  
Angelo Favini
Keyword(s):  
Hilbert Space ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Maximal Monotone ◽  
Time Discretization ◽  
Cauchy Problems ◽  
Initial Value ◽  
Selfadjoint Operators ◽  
Convergence Results ◽  
Uniqueness Of The Solution

In this paper we deal with the equationL(d2u/dt2)+B(du/dt)+Au∋f, whereLandAare linear positive selfadjoint operators in a Hilbert spaceHand from a Hilbert spaceV⊂Hto its dual spaceV′, respectively, andBis a maximal monotone operator fromVtoV′. By assuming some coerciveness onL+BandA, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.

On a quasilinear wave equation with nonlinear damping

1988 ◽  
Vol 110 (3-4) ◽  
pp. 227-239 ◽  
Author(s):  
Dang Dinh Hai
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Damping ◽  
Quasilinear Wave Equation ◽  
Uniqueness Of The Solution ◽  
Global Existence And Uniqueness ◽  
Image Position

SynopsisWe prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equationby using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.

scholarly journals Transforms on white noise functionals with their applications to Cauchy problems

1997 ◽  
Vol 147 ◽  
pp. 1-23 ◽  
Author(s):  
Dong Myung Chung ◽  
Un Cig Ji
Keyword(s):  
Cauchy Problem ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Characterization Theorem ◽  
Cauchy Problems ◽  
Continuous Linear ◽  
The Cauchy Problem ◽  
White Noise Functionals ◽  
Generalized Laplacian

AbstractA generalized Laplacian ΔG(K) is defined as a continuous linear operator acting on the space of test white noise functionals. Operator-parameter - and -transforms on white noise functionals are introduced and then prove a characterization theorem for and -transforms in terms of the coordinate differential operator and the coordinate multiplication. As an application, we investigate the existence and uniqueness of solution of the Cauchy problem for the heat equation associated with ΔG(K)

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