Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent
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2024
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| _version_ | 1866917766835994624 |
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| author | Lin, Jiayun Ikeda, Masahiro |
| author_facet | Lin, Jiayun Ikeda, Masahiro |
| contents | In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where $b(t)\approx (1+t)^{-β}$. In the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ by the framework of ordinary differential inequality. The blow-up results, with the global existence in the supercritical case $p_c<p<\frac{N}{N-σ}$ obtained in [19], shows that the critical exponent for the fractional wave quation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, together with the lower estimate of lifespan derived in [19], we could conclude that the estimate in this paper is sharp. Note that the our result of the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_10552 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent Lin, Jiayun Ikeda, Masahiro Analysis of PDEs 35B44, 35A01, 35L15, 35L05 In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where $b(t)\approx (1+t)^{-β}$. In the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ by the framework of ordinary differential inequality. The blow-up results, with the global existence in the supercritical case $p_c<p<\frac{N}{N-σ}$ obtained in [19], shows that the critical exponent for the fractional wave quation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, together with the lower estimate of lifespan derived in [19], we could conclude that the estimate in this paper is sharp. Note that the our result of the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan. |
| title | Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent |
| topic | Analysis of PDEs 35B44, 35A01, 35L15, 35L05 |
| url | https://arxiv.org/abs/2401.10552 |