On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911642491551744 |
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| author | Hamza, Mohamed Ali |
| author_facet | Hamza, Mohamed Ali |
| contents | This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\fracμ{1+t} \partial_t u$, the mass term $\frac{ν^2}{(1+t)^2} u$, and a time-derivative nonlinearity $| \partial_t u |^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $δ= (μ-1)^2 - 4ν^2$ is not a structural prerequisite for determining the blow-up range. Indeed, we show that even in the regime $δ< 0$, the blow-up region remains invariant and is uniquely determined by the shifted dimension $n+μ$, aligning with the Glassey-type critical exponent. Our result suggest that the classical restriction $δ\ge 0$ is due to a technical tool rather than an intrinsic feature of the blow-up mechanism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06478 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction Hamza, Mohamed Ali Analysis of PDEs 35L71, 35B44 This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\fracμ{1+t} \partial_t u$, the mass term $\frac{ν^2}{(1+t)^2} u$, and a time-derivative nonlinearity $| \partial_t u |^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $δ= (μ-1)^2 - 4ν^2$ is not a structural prerequisite for determining the blow-up range. Indeed, we show that even in the regime $δ< 0$, the blow-up region remains invariant and is uniquely determined by the shifted dimension $n+μ$, aligning with the Glassey-type critical exponent. Our result suggest that the classical restriction $δ\ge 0$ is due to a technical tool rather than an intrinsic feature of the blow-up mechanism. |
| title | On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction |
| topic | Analysis of PDEs 35L71, 35B44 |
| url | https://arxiv.org/abs/2604.06478 |