Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process

2004 ◽  
Vol 4 (2) ◽  
Author(s):  
Martin Ondrej�t
Keyword(s):  
Wiener Process ◽  
Evolution Equations ◽  
Strong Solutions ◽  
Spatially Homogeneous

scholarly journals ON UNIQUENESS OF MILD SOLUTIONS FOR DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 13 (03) ◽  
pp. 363-376 ◽  
Author(s):  
CARLO MARINELLI ◽  
MICHAEL RÖCKNER
Keyword(s):  
Evolution Equations ◽  
Broad Class ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Fundamental Issue ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Monotonicity Assumption ◽  
Semigroup Approach

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

scholarly journals Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Jung-Chan Chang ◽  
Hsiang Liu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Initial Conditions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Strong Solutions ◽  
Interpolation Spaces ◽  
Nonlocal Initial Conditions ◽  
Densely Defined ◽  
Semilinear Evolution Equations

This paper is concerned with the existence of mild and strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach spaceX. Considering the equations in the norm of some interpolation spaces betweenXand the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.

scholarly journals THE SINGULARITY IN GENERIC GRAVITATIONAL COLLAPSE IS SPACELIKE, LOCAL AND OSCILLATORY

1998 ◽  
Vol 13 (19) ◽  
pp. 1565-1573 ◽  
Author(s):  
B. K. BERGER ◽  
D. GARFINKLE ◽  
J. ISENBERG ◽  
V. MONCRIEF ◽  
M. WEAVER
Keyword(s):  
Numerical Simulations ◽  
Gravitational Collapse ◽  
Evolution Equations ◽  
Oscillatory Behavior ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Spatial Point ◽  
Spatially Inhomogeneous ◽  
Homogeneous Universe ◽  
Spatially Homogeneous

A longstanding conjecture by Belinskii, Khalatnikov and Lifshitz that the singularity in generic gravitational collapse is spacelike, local and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological space–times. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.

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