# Cauchy Problem (cauchy + problem)

Distribution by Scientific Domains

## Selected Abstracts

### Schrödinger equations of higher order

MATHEMATISCHE NACHRICHTEN, Issue 7 2007
Alessia Ascanelli
Abstract We are interested in finding the sharp regularity with respect to the time variable of the coefficients of a Schrödinger type operator in order to have a well-posed Cauchy Problem in H,. We consider both the cases of the first derivative that breaks down at a point t0 and of Log-Lipschitz coefficients. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Global well-posedness of the Cauchy problem for certain magnetohydrodynamic-, models

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010
Yi Du
Abstract This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics-, model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as ,,0, the MHD-, model reduces to the MHD equations, and the solutions of the MHD-, model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd. [source]

### Fragmentation arising from a distributional initial condition

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2010
W. Lamb
Abstract A standard model for pure fragmentation is subjected to an initial condition of Dirac-delta type. Results for a corresponding abstract Cauchy problem are derived via the theory of equicontinuous semigroups of operators on locally convex spaces. An explicit solution is obtained for the case of a power-law kernel. Rigorous justification is thereby provided for results obtained more formally by Ziff and McGrady. Copyright © 2010 John Wiley & Sons, Ltd. [source]

### Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein,Gordon equation with dissipative term

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2010
Xu Runzhang
Abstract We study the Cauchy problem of nonlinear Klein,Gordon equation with dissipative term. By introducing a family of potential wells, we derive the invariant sets and prove the global existence, finite time blow up as well as the asymptotic behaviour of solutions. In particular, we show a sharp condition for global existence and finite time blow up of solutions. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009
Yi Zhou
Abstract In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263,1317; Nonlinear Anal. 1997; 28:1299,1322; Chin. Ann. Math. 2004; 25B:37,56). We give a new, very simple proof of this result and also give a sharp point-wise decay estimate of the solution. Then, we consider the mixed initial-boundary-value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59,78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point-wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Pseudoparabolic equations with additive noise and applications

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2009
Abstract In this work we present some results on the Cauchy problem for a general class of linear pseudoparabolic equations with additive noise. We consider questions of existence and uniqueness of mild and strong solutions and well posedness for this problem. We also prove the existence and uniqueness of mild and strong solutions for a related perturbed Cauchy problem and we investigate the continuity of the solution with respect to a small parameter. The abstract results are illustrated using examples from electromagnetics and heat conduction. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### On the well-posedness of the Cauchy problem for an MHD system in Besov spaces

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2009
Changxing Miao
Abstract This paper is devoted to the study of the Cauchy problem of incompressible magneto-hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n,3, we establish the global well-posedness of the Cauchy problem of an incompressible magneto-hydrodynamics system for small data and the local one for large data in the Besov space , (,n), 1,p<, and 1,r,,. Meanwhile, we also prove the weak,strong uniqueness of solutions with data in , (,n),L2(,n) for n/2p+2/r>1. In the case of n=2, we establish the global well-posedness of solutions for large initial data in homogeneous Besov space , (,2) for 2[source]

### Global existence for a contact problem with adhesion

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2008
Elena Bonetti
Abstract In this paper, we analyze a contact problem with irreversible adhesion between a viscoelastic body and a rigid support. On the basis of Frémond's theory, we detail the derivation of the model and of the resulting partial differential equation system. Hence, we prove the existence of global in time solutions (to a suitable variational formulation) of the related Cauchy problem by means of an approximation procedure, combined with monotonicity and compactness tools, and with a prolongation argument. In fact the approximate problem (for which we prove a local well-posedness result) models a contact phenomenon in which the occurrence of repulsive dynamics is allowed for. We also show local uniqueness of the solutions, and a continuous dependence result under some additional assumptions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Global well posedness for the Gross,Pitaevskii equation with an angular momentum rotational term

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2008
Chengchun Hao
Abstract In this paper, we establish the global well posedness of the Cauchy problem for the Gross,Pitaevskii equation with a rotational angular momentum term in the space ,2. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Mechanism of the formation of singularities for quasilinear hyperbolic systems with linearly degenerate characteristic fields

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2008
Ta-Tsien Li
Abstract One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see Arch. Rational Mech. Anal. 2004; 172:65,91; Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer: New York, 1984) and it is still an open problem in the general situation up to now. In this paper, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are given. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2007
Ying Wang
Abstract In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the multidimensional generalized Boussinesq equation are obtained. Furthermore, the blow-up of the solution for the Cauchy problem of the generalized Boussinesq equation is proved. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### The asymptotic behaviour of global smooth solutions to the multi-dimensional hydrodynamic model for semiconductors

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2003
Ling Hsiao
Abstract We establish the global existence of smooth solutions to the Cauchy problem for the multi-dimensional hydrodynamic model for semiconductors, provided that the initial data are perturbations of a given stationary solutions, and prove that the resulting evolutionary solution converges asymptotically in time to the stationary solution exponentially fast. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Loss of regularity for the solutions to hyperbolic equations with non-regular coefficients,an application to Kirchhoff equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2003
Fumihiko Hirosawa
We consider the Cauchy problem for second-order strictly hyperbolic equations with time-depending non-regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C, well-posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### On L1 decay problem for the dissipative wave equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2003
Kosuke Ono
Abstract We study the decay estimates of solutions to the Cauchy problem for the dissipative wave equation in one, two, and three dimensions. The representation formulas of the solutions provide the sharp decay rates on L1 norms and also Lp norms. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### On singular mono-energetic transport equations in slab geometry

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2002
Mohamed Chabi
In this paper we establish the well posedness of the Cauchy problem associated to transport equations with singular cross-sections (i.e. unbounded collisions frequencies and unbounded collision operators) in L1 spaces for specular reflecting boundary conditions. In addition, we discuss the weak compactness of the second-order remainder term of the Dyson,Phillips expansion. This allows us to estimate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived. The case of singular transport equations with periodic boundary conditions is also discussed. The proofs make use of the Miyadera perturbation theory of positive semigroups on AL -spaces. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### A critical exponent in a degenerate parabolic equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2002
Michael Winkler
We consider positive solutions of the Cauchy problem in for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### Local smoothing for Kato potentials in three dimensions

MATHEMATISCHE NACHRICHTEN, Issue 10 2009
J. A. Barceló
Abstract We prove weighted local smoothing estimates for the resolvent of the Laplacian in three dimensions with weights belonging to the Kerman,Sawyer class. This class contains the well-known global Kato and Rollnik classes. We go on to discuss dispersive and Strichartz estimates for perturbations of the Laplacian by small potentials, and apply our results and observations to the well-posedness in L2 of the Cauchy problem for some linear and semilinear Schrödinger equations (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipation

MATHEMATISCHE NACHRICHTEN, Issue 12 2008
Nour-Eddine Amroun
Abstract In this paper we prove the existence of global decaying H2 solutions to the Cauchy problem for a wave equation with a nonlinear dissipative term by constructing a stable set in H1(,n). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### The Cauchy problem for quasilinear SG-hyperbolic systems

MATHEMATISCHE NACHRICHTEN, Issue 7 2007
Marco Cappiello
Abstract We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) , [0, T ] × ,n and presenting a linear growth for |x | , ,. We prove well-posedness in the Schwartz space ,, (,n). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

MATHEMATISCHE NACHRICHTEN, Issue 10 2005
Piero D'Ancona
Abstract We study the wellposedness in the Gevrey classes Gs and in C, of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### An iterative procedure for solving a Cauchy problem for second order elliptic equations

MATHEMATISCHE NACHRICHTEN, Issue 1 2004
Tomas JohanssonArticle first published online: 14 JUL 200
Abstract An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the Cauchy problem for second order strictly hyperbolic equations with non,regular coefficients

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
Fumihiko Hirosawa
Abstract In this paper we shall consider some necessary and sufficient conditions for well,posedness of second order hyperbolic equations with non,regular coefficients with respect to time. We will derive some optimal regularities for well,posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory. [source]

### An iterative method for the reconstruction of a stationary flow

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007
Tomas Johansson
Abstract In this article, an iterative algorithm based on the Landweber-Fridman method in combination with the boundary element method is developed for solving a Cauchy problem in linear hydrostatics Stokes flow of a slow viscous fluid. This is an iteration scheme where mixed well-posed problems for the stationary generalized Stokes system and its adjoint are solved in an alternating way. A convergence proof of this procedure is included and an efficient stopping criterion is employed. The numerical results confirm that the iterative method produces a convergent and stable numerical solution. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 [source]

### A Note on Analysis and Numerics for Radiative MHD in 3D

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Andreas Dedner
The equations of radiative magnetohydrodynamics describe the dynamics of an electrically conducting fluid interacting with magnetic fields and radiation. In particular they provide a widely accepted mathematical model for the physics in the solar photosphere and convection zone. In the spatially three-dimensional case we present several notions of solutions for the Cauchy problem and discuss existence and uniqueness issues. Furthermore we report on numerical experiments in the context of solar physics. [source]

### Exponential growth for the wave equation with compact time-periodic positive potential

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 4 2009
Ferruccio Colombini
We prove the existence of smooth positive potentials V(t, x), periodic in time and with compact support in x, for which the Cauchy problem for the wave equation utt , ,xu + V(t, x)u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z , ,, |z| > 1, associated to V(t, x). © 2008 Wiley Periodicals, Inc. [source]

### Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2006
Slim Ibrahim
We prove the existence and uniqueness of global solutions for a Cauchy problem associated to a semilinear Klein-Gordon equation in two space dimensions. Our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method. © 2006 Wiley Periodicals, Inc. [source]

### Equivariant wave maps in two space dimensions,

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2003
Michael Struwe
Singularities of corotational wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from ,,2 into ,. In consequence, for noncompact target surfaces of revolution, the Cauchy problem for wave maps is globally well-posed. © 2003 Wiley Periodicals, Inc. [source]

### Global well-posedness for two modified-Leray-,-MHD models with partial viscous terms

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2010
Yong Zhou
Abstract In this paper, we will prove global well-posedness for the Cauchy problems of two modified-Leray-,-MHD models with partial viscous terms. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Does the loss of regularity really appear?

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2009
Xiaojun Lu
Abstract In the theory of weakly hyperbolic equations we have the effect of loss of regularity. The present paper devotes to the study of two problems. On the one hand we describe families of weakly hyperbolic Cauchy problems for which we have no loss of regularity. On the other hand we discuss the question if the loss of derivatives really appears. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### L1 Decay estimates for dissipative wave equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001
Albert Milani
Let u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equation $$u_{tt}+u_t-\Delta u=0$$\nopagenumbers\end(1) and the heat equation $$v_t-\Delta v=0$$\nopagenumbers\end(2) We show that, as $t\rightarrow+\infty$\nopagenumbers\end, the norms $\|\partial_t^k\,D_x^\alpha u(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end and $\|\partial_t^k\,D_x^\alpha v(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end decay to 0 with the same polynomial rate. This result, which is well known for decay rates in $L^p({\rm R}^n)$\nopagenumbers\end with $2\leq p\leq+\infty$\nopagenumbers\end, provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1). Copyright © 2001 John Wiley & Sons, Ltd. [source]