Decay estimates for beam equations with potentials on the line
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910933989720064 |
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| author | Chen, Shuangshuang Wan, Zijun Yao, Xiaohua |
| author_facet | Chen, Shuangshuang Wan, Zijun Yao, Xiaohua |
| contents | This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$
\partial_t^2 u + \left( Δ^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$.
Let $H = Δ^2 + V$ and $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum of $H$. Then for $m = 0$, we establish the following decay estimates of the solution operators:
$$
\left\|\cos (t \sqrt{H}) P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} +
\left\|\frac{\sin (t \sqrt{H})}{t \sqrt{H}} P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} \lesssim |t|^{-\frac{1}{2}}.
$$ But for $m \neq 0$, the solutions have different time decay estimates from the case where $m=0$. Specifically, the $L^1$-$L^\infty$ estimates of $\cos (t \sqrt{H + m^2})$ and $\frac{\sin (t \sqrt{H + m^2})}{\sqrt{H + m^2}}$ are bounded by $O(|t|^{-\frac{1}{4}})$ in the low-energy part and $O(|t|^{-\frac{1}{2}})$ in the high-energy part.
It is noteworthy that all these results remain consistent with the free cases (i.e., $V = 0$) whatever zero is a regular point or a resonance of $H$. As consequences, we establish the corresponding Strichartz estimates, which are fundamental to study nonlinear problems of beam equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decay estimates for beam equations with potentials on the line Chen, Shuangshuang Wan, Zijun Yao, Xiaohua Analysis of PDEs This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( Δ^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = Δ^2 + V$ and $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum of $H$. Then for $m = 0$, we establish the following decay estimates of the solution operators: $$ \left\|\cos (t \sqrt{H}) P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} + \left\|\frac{\sin (t \sqrt{H})}{t \sqrt{H}} P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} \lesssim |t|^{-\frac{1}{2}}. $$ But for $m \neq 0$, the solutions have different time decay estimates from the case where $m=0$. Specifically, the $L^1$-$L^\infty$ estimates of $\cos (t \sqrt{H + m^2})$ and $\frac{\sin (t \sqrt{H + m^2})}{\sqrt{H + m^2}}$ are bounded by $O(|t|^{-\frac{1}{4}})$ in the low-energy part and $O(|t|^{-\frac{1}{2}})$ in the high-energy part. It is noteworthy that all these results remain consistent with the free cases (i.e., $V = 0$) whatever zero is a regular point or a resonance of $H$. As consequences, we establish the corresponding Strichartz estimates, which are fundamental to study nonlinear problems of beam equations. |
| title | Decay estimates for beam equations with potentials on the line |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.09061 |