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Register a new userOn the defocusing semilinear wave equations in three space dimension with small power
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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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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.
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 $p>2sqrt{5}-1$.
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.
It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent, see Ikehata-Todorova-Yordanov cite{ITY}(the bottom in page 2) and Nishihara-Sobajima-Wakasugi cite{N2}(conjecture iii in page 4). In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the lqlq hyperbolicity of the equation.
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.
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