Decay characterization of solutions to semi-linear structurally damped $σ$-evolution equations with time-dependent damping
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913308534112256 |
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| author | Anh, Cung The An, Phan Duc Duong, Pham Trieu |
| author_facet | Anh, Cung The An, Phan Duc Duong, Pham Trieu |
| contents | In this paper, we study the Cauchy problem to the linear damped $σ$-evolution equation with time-dependent damping in the effective cases \begin{equation*} u_{t t}+(-Δ)^σu+b(t)(-Δ)^δu_t=0, \end{equation*} and investigate the decay rates of the solution and its derivatives that are expressed in terms of the decay character of the initial data $u_0(x)=u(0, x)$ and $u_1(x)=u_t(0, x)$. We are interested also in the existence and decay rate of the global in time solution with small data for the corresponding semi-linear problem with the nonlinear term of power type $||D|^γu|^p$. The blow-up results for solutions to the semi-linear problem in the case $γ=0$ are presented to show the sharpness of the exponent $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06855 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decay characterization of solutions to semi-linear structurally damped $σ$-evolution equations with time-dependent damping Anh, Cung The An, Phan Duc Duong, Pham Trieu Analysis of PDEs 35A01, 35G25, 35L71 In this paper, we study the Cauchy problem to the linear damped $σ$-evolution equation with time-dependent damping in the effective cases \begin{equation*} u_{t t}+(-Δ)^σu+b(t)(-Δ)^δu_t=0, \end{equation*} and investigate the decay rates of the solution and its derivatives that are expressed in terms of the decay character of the initial data $u_0(x)=u(0, x)$ and $u_1(x)=u_t(0, x)$. We are interested also in the existence and decay rate of the global in time solution with small data for the corresponding semi-linear problem with the nonlinear term of power type $||D|^γu|^p$. The blow-up results for solutions to the semi-linear problem in the case $γ=0$ are presented to show the sharpness of the exponent $p$. |
| title | Decay characterization of solutions to semi-linear structurally damped $σ$-evolution equations with time-dependent damping |
| topic | Analysis of PDEs 35A01, 35G25, 35L71 |
| url | https://arxiv.org/abs/2404.06855 |