Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity
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arXiv
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2025
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| author | Fino, Ahmad Z. Hamza, Mohamed Ali |
| author_facet | Fino, Ahmad Z. Hamza, Mohamed Ali |
| contents | We investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $μ_0(1+x^2)^{-1/2}$, where $μ_0>0$, and time derivative nonlinearity.
We establish global existence of mild solutions for small data compactly supported by employing energy estimates within suitable Sobolev spaces of the associated homogeneous problem.
Furthermore, we derive a blow-up result under some positive initial data by employing the test function method. This shows that the critical exponent is given by $p_G(1+μ_0)=1+2/μ_0$, when $μ_0\in (0,1]$, where $p_G$ is the Glassey exponent. To the best of our knowledge, this constitutes the first identification of the critical exponent range for this class of equations. As by product, we extend the global existence result to a more general class of space/time nonlinearities of the form $f(\partial_tu,\partial_x u)=|\partial_x u|^{q}$ or $f(\partial_tu,\partial_x u)=|\partial_tu|^{p}|\partial_x u|^{q}$, with $p,q>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity Fino, Ahmad Z. Hamza, Mohamed Ali Analysis of PDEs 35A01, 35B33, 35L15, 35D35, 35B44 We investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $μ_0(1+x^2)^{-1/2}$, where $μ_0>0$, and time derivative nonlinearity. We establish global existence of mild solutions for small data compactly supported by employing energy estimates within suitable Sobolev spaces of the associated homogeneous problem. Furthermore, we derive a blow-up result under some positive initial data by employing the test function method. This shows that the critical exponent is given by $p_G(1+μ_0)=1+2/μ_0$, when $μ_0\in (0,1]$, where $p_G$ is the Glassey exponent. To the best of our knowledge, this constitutes the first identification of the critical exponent range for this class of equations. As by product, we extend the global existence result to a more general class of space/time nonlinearities of the form $f(\partial_tu,\partial_x u)=|\partial_x u|^{q}$ or $f(\partial_tu,\partial_x u)=|\partial_tu|^{p}|\partial_x u|^{q}$, with $p,q>1$. |
| title | Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity |
| topic | Analysis of PDEs 35A01, 35B33, 35L15, 35D35, 35B44 |
| url | https://arxiv.org/abs/2502.07313 |