Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces [PDF]
, 2018 For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity.
Ehrnström, Mats, Pei, Long
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The low regularity global solutions for the critical generalized KdV equation [PDF]
, 2009 We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case.
Miao, Changxing +3 more
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Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces [PDF]
, 2010 Let $F(u)=h$ be an operator equation in a Banach space $X$, $\|F'(u)-F'(v)\|\leq \omega(\|u-v\|)$, where $\omega\in C([0,\infty))$, $\omega(0)=0$, $\omega(r)>0$ if $r>0$, $\omega(r)$ is strictly growing on $[0,\infty)$. Denote $A(u):=F'(u)$, where $F'(u)$
Ramm, A. G.
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Asymptotic stability of solutions to abstract differential equations [PDF]
, 2010 An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H ...
Ramm, A. G.
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Mean Field Asymptotic Behavior of Quantum Particles with Initial Correlations
, 2014 In the paper we consider the problem of the rigorous description of the kinetic evolution in the presence of initial correlations of quantum large particle systems.
Gerasimenko, V. I.
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A nonlinear inequality and evolution problems [PDF]
, 2010 Assume that $g(t)\geq 0$, and $$\dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt}, $$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right, $\dot{g}(t):=\lim_{s ...
Ramm, A. G.
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Scattering below ground state of 3D focusing cubic fractional Schordinger equation with radial data
, 2017 The aim of this note is to adapt the strategy in [4][See,B.Dodson, J.Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for ...
Sun, Chenmin +3 more
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Stability of solutions to some evolution problem [PDF]
, 2010 Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $t\in \R_+:=[0,\
Ramm, A. G.
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Trait evolution in two-sex populations
, 2015 We present an individual-based model of phenotypic trait evolution in two-sex populations, which includes semi-random mating of individuals of the opposite sex, natural death and intra-specific competition.
Zwoleński, Paweł
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Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential
, 2019 We study the uniform resolvent estimates for Schr\"odinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-\Delta+V(x)$ where $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(\theta) r^{-2}$ where $r=|x|, \theta=x/|x|$
Mizutani, Haruya +2 more
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