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Register a new userPointwise decay for semilinear wave equations in $mathbb{R}^{!+3}$
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Shiwu Yang
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In this paper, we use Dafermos-Rodnianskis new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in $mathbb{R}^{1+3}$. We prove that the solution decays as quickly as linear waves for $p>frac{1+sqrt{17}}{2}$, covering part of the sub-conformal case, while for the range $2<pleq frac{1+sqrt{17}}{2}$, the solution still decays with rate at least $t^{-frac{1}{3}}$. As a consequence, the solution scatters in energy space when $p>2.3542$. We also show that the solution is uniformly bounded when $p>frac{3}{2}$.
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This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.
We prove that solution of defocusing semilinear wave equation in $mathbb{R}^{1+3}$ with pure power nonlinearity is uniformly bounded for all $frac{3}{2}<pleq 2$ with sufficiently smooth and localized data. The result relies on the $r$-weighted energy estimate originally introduced by Dafermos and Rodnianski. This appears to be the first result regarding the global asymptotic property for the solution with small power $p$ under 2.
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities begin{equation*} -left(a+bint_{mathbb{R}^{3}}| abla u(x)|^{2}right) Delta u =lambda u+|u|^{p-2}u+u^{5}quad text{for some} lambdainmathbb{R},quad xinmathbb{R}^{3}, end{equation*} with prescribed $L^{2}$-norm mass begin{equation*} int_{mathbb{R}^{3}}u^{2}=c^{2} end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},lambda_{c})in S(c)times mathbb{R}$ for any $c>0$, $a,b >0$ and $frac{14}{3}leq p< 6,$ where $S(c)={uin H^{1}(mathbb{R}^{3}):int_{mathbb{R}^{3}}u^{2}=c^{2}}.$ textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states oindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}
In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+sqrt{8}$. Moreover combined with Br{e}zis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-frac{1}{2}}$ when $p>frac{11}{3}$ and with rate $t^{ -frac{p-1}{8}+epsilon }$ for all $1<pleq frac{11}{3}$. This in particular implies that the solution scatters in energy space when $p>2sqrt{5}-1$.
We prove global well-posedness for the $3D$ radial defocusing cubic wave equation with data in $H^{s} times H^{s-1}$, $1>s>{7/10}$.
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