Authors:Caroline Bauzet, Vincent Castel, Julia Charrier Pages: 213 - 294 Abstract: Journal of Hyperbolic Differential Equations, Volume 17, Issue 02, Page 213-294, June 2020. We are interested in multi-dimensional nonlinear scalar conservation laws forced by a multiplicative stochastic noise with a general time and space dependent flux-function. We address simultaneously theoretical and numerical issues in a general framework and consider a larger class of flux functions in comparison to the one in the literature. We establish existence and uniqueness of a stochastic entropy solution together with the convergence of a finite volume scheme. The novelty of this paper is the use of a numerical approximation (instead of a viscous one) in order to get, both, the existence and the uniqueness of solutions. The quantitative bounds in our uniqueness result constitute a preliminary step toward the establishment of strong error estimates. We also provide an [math] stability result for the stochastic entropy solution. Citation: Journal of Hyperbolic Differential Equations PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0219891620500071 Issue No:Vol. 17, No. 02 (2020)
Authors:Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto Pages: 295 - 354 Abstract: Journal of Hyperbolic Differential Equations, Volume 17, Issue 02, Page 295-354, June 2020. We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein–Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space [math]. We prove that any energy solution has a global bound of the [math] space-time norm, and hence, scatters in [math] as [math]. The proof is based on the argument by Killip–Stovall–Visan (Trans. Amer. Math. Soc. 364(3) (2012) 1571–1631). However, since well-posedness in [math] for NLKG with the exponential nonlinearity holds only for small initial data, we use the [math]-norm for some [math] instead of the [math]-norm, where [math] denotes the [math]th order [math]-based Sobolev space. Citation: Journal of Hyperbolic Differential Equations PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0219891620500083 Issue No:Vol. 17, No. 02 (2020)
Authors:Luigi Forcella, Lysianne Hari Pages: 355 - 394 Abstract: Journal of Hyperbolic Differential Equations, Volume 17, Issue 02, Page 355-394, June 2020. We consider the pure-power defocusing nonlinear Klein–Gordon equation, in the [math]-subcritical case, posed on the product space [math], where [math] is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space [math] for [math]. The strategy consists in proving a suitable profile decomposition theorem on the whole manifold to pursue a concentration-compactness and rigidity method along with the proofs of (global in time) Strichartz estimates. Citation: Journal of Hyperbolic Differential Equations PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0219891620500095 Issue No:Vol. 17, No. 02 (2020)
Authors:Halit Sevki Aslan, Michael Reissig Pages: 395 - 442 Abstract: Journal of Hyperbolic Differential Equations, Volume 17, Issue 02, Page 395-442, June 2020. We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity. Citation: Journal of Hyperbolic Differential Equations PubDate: 2020-08-07T07:00:00Z DOI: 10.1142/S0219891620500101 Issue No:Vol. 17, No. 02 (2020)
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