Improvement of conditions for finite time blow-up in a fourth-order nonlocal parabolic equation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914915158065152 |
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| author | Meng, Jingbo Qiu, Shuyan Xu, Guangyu Yi, Hong |
| author_facet | Meng, Jingbo Qiu, Shuyan Xu, Guangyu Yi, Hong |
| contents | This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition,
\begin{equation*}
\left\{\begin{array}{ll}\ds u_{t}+u_{xxxx}=|u|^{p-1}u-\frac{1}{a}\int_{0}^a|u|^{p-1}u\ dx, &
u_x(0)=u_x(a)=u_{xxx}(0)=u_{xxx}(a)=0, &
u(x,0)=u_0(x)\in H^2(0, a),\ \ \int_0^au_0(x)\ dx=0, &\end{array}\right.
\end{equation*}
where $a$ is a positive constant and $p>1$. The existing results on the problem suggest that the weak solution will blow up in finite time if $I(u_0)<0$ and the initial energy satisfies some appropriate assumptions, here $I(u_0)$ is the initial Nehari functional. In this paper, we extend the previous blow-up conditions with proving that those assumptions on the energy functional are superfluous and only $I(u_0)<0$ is sufficient to ensure the weak solution blowing up in finite time. Our conclusion depicts the significant influence of mass conservation on the dynamic behavior of solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improvement of conditions for finite time blow-up in a fourth-order nonlocal parabolic equation Meng, Jingbo Qiu, Shuyan Xu, Guangyu Yi, Hong Analysis of PDEs This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition, \begin{equation*} \left\{\begin{array}{ll}\ds u_{t}+u_{xxxx}=|u|^{p-1}u-\frac{1}{a}\int_{0}^a|u|^{p-1}u\ dx, & u_x(0)=u_x(a)=u_{xxx}(0)=u_{xxx}(a)=0, & u(x,0)=u_0(x)\in H^2(0, a),\ \ \int_0^au_0(x)\ dx=0, &\end{array}\right. \end{equation*} where $a$ is a positive constant and $p>1$. The existing results on the problem suggest that the weak solution will blow up in finite time if $I(u_0)<0$ and the initial energy satisfies some appropriate assumptions, here $I(u_0)$ is the initial Nehari functional. In this paper, we extend the previous blow-up conditions with proving that those assumptions on the energy functional are superfluous and only $I(u_0)<0$ is sufficient to ensure the weak solution blowing up in finite time. Our conclusion depicts the significant influence of mass conservation on the dynamic behavior of solution. |
| title | Improvement of conditions for finite time blow-up in a fourth-order nonlocal parabolic equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.09190 |