Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
Yong Chen Hongjun Gao María J. Garrido–Atienza Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 79-98 doi: 10.3934/dcds.2014.34.79
This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
keywords: Stochastic PDEs fractional Brownian motion pathwise solutions random dynamical systems.
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen Hongjun Gao Yue Liu
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3407-3441 doi: 10.3934/dcds.2013.33.3407
Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
keywords: Two-component Dullin-Gottwald-Holm system global solutions regularization wave-breaking solitary-wave solutions.

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