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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.13353 |
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| _version_ | 1866911006799691776 |
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| author | Fino, Ahmad Z. Hamza, Mohamed Ali |
| author_facet | Fino, Ahmad Z. Hamza, Mohamed Ali |
| contents | This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- Δu +\fracμ{1+t}\partial_tu= f(\partial_tu), \quad x\in \mathbb{R}^n, t>0,$$ where $f(\partial_tu)=|\partial_tu|^p $ or $|\partial_tu|^{p-1}\partial_tu$ with $p>1$ and $μ>0$. We prove global existence of small data solutions in low dimensions $1\leq n\leq 3$ by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for $p>1+\frac2μ$, in the one-dimensional case, when $μ\le 2$, which in conjunction with prior blow-up results from \cite{Our2}, establish that the critical exponent for small data solutions in one dimension is $p_G(1,μ)=1+\frac2μ$, when $μ\le 2$. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity Fino, Ahmad Z. Hamza, Mohamed Ali Analysis of PDEs 35A01, 35B33, 35L15, 35D35 This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- Δu +\fracμ{1+t}\partial_tu= f(\partial_tu), \quad x\in \mathbb{R}^n, t>0,$$ where $f(\partial_tu)=|\partial_tu|^p $ or $|\partial_tu|^{p-1}\partial_tu$ with $p>1$ and $μ>0$. We prove global existence of small data solutions in low dimensions $1\leq n\leq 3$ by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for $p>1+\frac2μ$, in the one-dimensional case, when $μ\le 2$, which in conjunction with prior blow-up results from \cite{Our2}, establish that the critical exponent for small data solutions in one dimension is $p_G(1,μ)=1+\frac2μ$, when $μ\le 2$. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity. |
| title | Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity |
| topic | Analysis of PDEs 35A01, 35B33, 35L15, 35D35 |
| url | https://arxiv.org/abs/2409.13353 |