9,739 research outputs found
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations
This study deals with the analysis of the Cauchy problem of a general class
of nonlocal nonlinear equations modeling the bi-directional propagation of
dispersive waves in various contexts. The nonlocal nature of the problem is
reflected by two different elliptic pseudodifferential operators acting on
linear and nonlinear functions of the dependent variable, respectively. The
well-known doubly dispersive nonlinear wave equation that incorporates two
types of dispersive effects originated from two different dispersion operators
falls into the category studied here. The class of nonlocal nonlinear wave
equations also covers a variety of well-known wave equations such as various
forms of the Boussinesq equation. Local existence of solutions of the Cauchy
problem with initial data in suitable Sobolev spaces is proven and the
conditions for global existence and finite-time blow-up of solutions are
established.Comment: 17 page
On the 2d Zakharov system with L^2 Schr\"odinger data
We prove local in time well-posedness for the Zakharov system in two space
dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the
space of optimal regularity in the sense that the data-to-solution map fails to
be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev
scale. Moreover, it is a natural space for the Cauchy problem in view of the
subsonic limit equation, namely the focusing cubic nonlinear Schroedinger
equation. The existence time we obtain depends only upon the corresponding
norms of the initial data - a result which is false for the cubic nonlinear
Schroedinger equation in dimension two - and it is optimal because
Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change
A mixed nonlinear time-fractional Rayleigh-Stokes equation
. This paper investigates a nonlinear time-fractional Rayleigh-Stokes
equation with mixed nonlinearities containing a power-type function, a logarithmic function and an inverse time-forcing term. Applying Lagrange’s mean
value theorem and the compactness of the Sobolev embeddings, we estimate
the complex Lipschitz property of mixed nonlinearity. We investigate the local
well-posed results (local existence, regularity estimate, continuation) of the solutions in Hilbert scales space. Moreover, the global existence theory affiliated
to the finite-time blow-up is considere
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