Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the decay properties of solutions to a class of Schrodinger equations
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
