Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
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| Format: | Preprint |
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2023
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| _version_ | 1866916082557648896 |
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| author | Floridia, Giuseppe Liu, Yikan Yamamoto, Masahiro |
| author_facet | Floridia, Giuseppe Liu, Yikan Yamamoto, Masahiro |
| contents | This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_12724 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms Floridia, Giuseppe Liu, Yikan Yamamoto, Masahiro Analysis of PDEs 35R11, 35K58, 35B44 This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem. |
| title | Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms |
| topic | Analysis of PDEs 35R11, 35K58, 35B44 |
| url | https://arxiv.org/abs/2302.12724 |