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Global Existence (global + existence)

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Selected Abstracts

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]

Analysis of a moving boundary value problem arising in biofilm modelling

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2008
Barbara Szomolay
Abstract We consider a moving boundary value problem associated with a 1-D biofilm model proposed by Szomolay et al. (Environ. Microbiol. 2005; 8:1186,1191). The new model includes growth and detachment which make it more realistic in a biofilm setting. Global existence and properties of solutions are shown using the method of characteristics. We also study the existence of the corresponding steady-state solutions and prove their uniqueness for small doses of biocide. In addition, sufficient conditions for the existence of trivial/nontrivial steady states are established. Copyright © 2008 John Wiley & Sons, Ltd. [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 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]

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007
Salim A. Messaoudi
Abstract In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003
J. Gawinecki
Abstract We consider some initial,boundary value problems for non-linear equations of thermoviscoelasticity in the three-dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Global existence for the Vlasov,Darwin system in ,3 for small initial data

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2003
Saïd Benachour
We prove the global existence of weak solutions to the Vlasov,Darwin system in R3 for small initial data. The Vlasov,Darwin system is an approximation of the Vlasov,Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov,Maxwell system, the total energy conservation does not provide an L2-bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov,Darwin system towards the Vlasov,Poisson system. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2002
Krzysztof Che
This article introduces the notion of weak-type solutions for systems of equations from the theory of inelastic deformations, assuming that the considered model is of monotone type (for the definition see [Lecture Notes in Mathematics, 1998, vol. 1682]). For the boundary data associated with the initial-boundary value problem and satisfying the safe-load condition the existence of global in time weak-type solutions is proved assuming that the monotone model is rate-independent or of gradient type. Moreover, for models possessing an additional regularity property (see Section 5) the existence of global solutions in the sense of measures, defined by Temam in Archives for Rational Mechanics and Analysis, 95: 137, is obtained, too. Copyright © 2002 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]

Global weak solution to the flow of liquid crystals system

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2009
Fei Jiang
Abstract In this paper, we study a simplified system for the flow of nematic liquid crystals in a bounded domain in the three-dimensional space. We derive the basic energy law which enables us to prove the global existence of the weak solutions under the condition that the initial density belongs to L,(,) for any . Especially, we also obtain that the weak solutions satisfy the energy inequality in integral or differential form. 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]

On the global existence and small dispersion limit for a class of complex Ginzburg,Landau equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2009
Hongjun Gao
Abstract In this paper we consider a class of complex Ginzburg,Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial-value problem in d -dimensional torus ,,d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd. [source]

BV-estimates of Lax,Friedrichs' scheme for hyperbolic conservation laws with relaxation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2008
Jing Zhang
Abstract In this paper, we will give BV-estimates of Lax,Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ,>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd. [source]

A fourth-order parabolic equation in two space dimensions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007
Changchun Liu
Abstract In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the Lp type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007
Salim A. Messaoudi
Abstract In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Blow up, decay bounds and continuous dependence inequalities for a class of quasilinear parabolic problems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2006
L. E. Payne
Abstract This paper deals with a class of semilinear parabolic problems. In particular, we establish conditions on the data sufficient to guarantee blow up of solution at some finite time, as well as conditions which will insure that the solution exists for all time with exponential decay of the solution and its spatial derivatives. In the case of global existence, we also investigate the continuous dependence of the solution with respect to some data of the problem. Copyright © 2005 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]

Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003
J. Gawinecki
Abstract We consider some initial,boundary value problems for non-linear equations of thermoviscoelasticity in the three-dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Global and blow-up solutions for non-linear degenerate parabolic systems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003
Zhi-wen Duan
Abstract In this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (,1=,2) of solution are analysed. For , the blow-up time, blow-up rate and blow-up set of blow-up solution are estimated and the asymptotic behaviour of solution near the blow-up time is discussed by using the ,energy' method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Global existence for the Vlasov,Darwin system in ,3 for small initial data

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2003
Saïd Benachour
We prove the global existence of weak solutions to the Vlasov,Darwin system in R3 for small initial data. The Vlasov,Darwin system is an approximation of the Vlasov,Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov,Maxwell system, the total energy conservation does not provide an L2-bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov,Darwin system towards the Vlasov,Poisson system. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On the standing wave in coupled non-linear Klein,Gordon equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2003
Jian Zhang
This paper is concerned with the standing wave in coupled non-linear Klein,Gordon equations. By an intricate variational argument we establish the existence of standing wave with the ground state. Then we derive out the sharp criterion for blowing up and global existence by applying the potential well argument and the concavity method. We also show the instability of the standing wave. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Thermoelasticity with second sound,exponential stability in linear and non-linear 1-d

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2002
Reinhard Racke
We consider linear and non-linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave-like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic,parabolic thermoelasticity is given. For Dirichlet type boundary conditions,rigidly clamped, constant temperature,the global existence of small, smooth solutions and the exponential stability are proved for a non-linear system. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Delayed quasilinear evolution equations with application to heat flow

MATHEMATISCHE NACHRICHTEN, Issue 5 2010
BártaArticle first published online: 15 MAR 2010
Abstract In this paper we show local and global existence for a class of (hyperbolic) quasilinear equations perturbed by bounded delay operators. In the last section, the abstract results are applied to a heat conduction model (© 2010 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]

Infinite time aggregation for the critical Patlak-Keller-Segel model in ,2

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2008
Adrien Blanchet
We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space ,2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local-in-time existence for any mass of "free-energy solutions," namely weak solutions with some free-energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8,/,. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t , , when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t , , if initially bounded. © 2007 Wiley Periodicals, Inc. [source]

Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2007
D. Bambusi
This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the Laplacian perturbed by a potential on Zoll manifolds. © 2007 Wiley Periodicals, Inc. [source]

Mean curvature flows and isotopy of maps between spheres

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2004
Mao-Pei Tsui
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is isotopic to a constant map. © 2004 Wiley Periodicals, Inc. [source]