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Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems

Yong-Fu Yang (2012)

Applications of Mathematics

In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant { ( t , x ) : t ≥ 0 , x ≥ 0 } is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a C 1 solution and its L 1 stability with certain small initial and boundary data.

Global controllability and stabilization for the nonlinear Schrödinger equation on an interval

Camille Laurent (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.

Global existence and energy decay of solutions to a Bresse system with delay terms

Abbes Benaissa, Mostefa Miloudi, Mokhtar Mokhtari (2015)

Commentationes Mathematicae Universitatis Carolinae

We consider the Bresse system in bounded domain with delay terms in the internal feedbacks and prove the global existence of its solutions in Sobolev spaces by means of semigroup theory under a condition between the weight of the delay terms in the feedbacks and the weight of the terms without delay. Furthermore, we study the asymptotic behavior of solutions using multiplier method.

Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping

Abderrahmane Zaraï, Nasser-eddine Tatar (2010)

Archivum Mathematicum

A viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping is considered. Using integral inequalities and multiplier techniques we establish polynomial decay estimates for the energy of the problem. The results obtained in this paper extend previous results by Tatar and Zaraï [25].

Global existence of solutions for a strongly coupled population system

Gonzalo Galiano, Ansgar Jüngel (2003)

Banach Center Publications

A strongly coupled cross-diffusion model for two competing species in a heterogeneous environment is analyzed. We sketch the proof of an existence result for the evolution problem with non-flux boundary conditions in one space dimension, completing previous results [4]. The proof is based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional.

Global existence of solutions for the 1-D radiative and reactive viscous gas dynamics

Wen Zhang, Jianwen Zhang (2012)

Applications of Mathematics

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.

Global existence of solutions to Schrödinger equations on compact riemannian manifolds below H 1

Sijia Zhong (2010)

Bulletin de la Société Mathématique de France

In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. s < 1 , under some bilinear Strichartz assumption. We will find some s ˜ < 1 , such that the solution is global for s > s ˜ .

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Elisabetta Rocca, Riccarda Rossi (2008)

Applications of Mathematics

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ , an evolution equation for the phase change parameter χ , including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable 𝐮 . The main novelty of the model...

Global solution to the Cauchy problem of nonlinear thermodiffusion in a solid body

Arkadiusz Szymaniec (2010)

Applicationes Mathematicae

We consider the initial-value problem for a nonlinear hyperbolic-parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove global (in time) existence and uniqueness of the solution to the initial-value problem for this nonlinear system. The global existence is proved using time decay estimates for the solution of the associated linearized problem. Next, we prove an energy estimate...

Global solutions of quasilinear systems of Klein–Gordon equations in 3D

Alexandru D. Ionescu, Benoît Pausader (2014)

Journal of the European Mathematical Society

We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.

Global time estimates for solutions to equations of dissipative type

Michael Ruzhansky, James Smith (2005)

Journées Équations aux dérivées partielles

Global time estimates of L p - L q norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.

Gradient flows with metric and differentiable structures, and applications to the Wasserstein space

Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second...

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