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Register a new userOn the decay properties of solutions to a class of Schrodinger equations
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We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections P+ and P- onto the positive and negative frequencies.
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In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrodinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_{th}$ and $ a {bf u}$, especially we show that $|u_{th}(r,z)|leq cleft(f{log r}{r}right)^{f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $om_{th}$, integral representations of ${bf u}$ in terms of $bm{om}=textit{curl }{bf u}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $( a u_r, a u_z)$ and $(om_r, om_{z})$, where $bm{om}= om_r {bf e}_r + om_{th} {bf e}_{th}+ om_z {bf e}_z$. Another is the first decay rate estimates in the $Oz$-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.
In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{itphi(sqrt{-Delta})}$, where $phi: mathbb{R}^+to mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.
This paper is devoted to prove the existence and nonexistence of positive solutions for a class of fractional Schrodinger equation in RN of the We apply a new methods to obtain the existence of positive solutions when f(u) is asymptotically linear with respect to u at infinity.
In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation begin{equation}label{1} onumber - Bigl(a+bint_{{R^3}} {{{left| { abla u} right|}^2}}Bigl) Delta u =lambda u+ {| u |^{p - 2}}u+mu {| u |^{q - 2}}u text { in } mathbb{R}^{3} end{equation} under the normalized constraint $int_{{mathbb{R}^3}} {{u}^2}=c^2$, where $a!>!0$, $b!>!0$, $c!>!0$, $2!<!q!<!frac{14}{3}!<! p!leq!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, $mu!>!0$ and $lambda!in!R$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p!=!6$ is involved and the corresponding energy functional is unbounded from below on $S_c=Big{ u in H^{1}({mathbb{R}^3}): int_{{mathbb{R}^3}} {{u}^2}=c^2 Big}$. If $2!<!q!<!frac{10}{3}$ and $frac{14}{3}!<! p!<!6$, we obtain a multiplicity result to the equation. If $2!<!q!<!frac{10}{3}!<! p!=!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of N. Soave (J. Differential Equations 2020 $&$ J. Funct. Anal. 2020), which studied the nonlinear Schr{o}dinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({int_{{R^3}} {left| { abla u} right|} ^2}) Delta u$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.
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