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Register a new userAsymptotic decay for defocusing semilinear wave equations in $mathbb{R}^{1+1}$
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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.
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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 $p>2sqrt{5}-1$.
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 $p>frac{3}{2}$.
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.
In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $mathbb{R}^{1+d}$ with $dgeq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space for $p$ with an improved lower bound. This in particular extends the existing scattering results to higher dimensions without spherical symmetry.
By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $mathbb{R}^{1+3}$ with pure power nonlinearity for all $1<pleq 2$. Consequently, the solution vanishes on the future null infinity and decays in time polynomially for all $sqrt{2}<pleq 2$. This improves the uniform boundedness result of the second author when $frac{3}{2}<pleq 2$.
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