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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.21769 |
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| _version_ | 1866913175554752512 |
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| author | Marcon, Diego Nascimento, Wanderley Santos, Matheus |
| author_facet | Marcon, Diego Nascimento, Wanderley Santos, Matheus |
| contents | We investigate the finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential and power nonlinearities in the oscillatory regime. For smooth, compactly supported, nonnegative initial data, we prove nonexistence of global-in-time solutions when the power nonlinearity lies below the positive root of an explicit Strauss-type polynomial naturally associated with the equation. The proof combines two main ingredients. The first is the construction of a positive adjoint temporal profile, which yields a weighted monotonicity formula and, consequently, a quantitative lower bound for the nonlinear term. The second is a phase-localized test function argument on logarithmic time shells, fitted to capture the oscillatory effects induced by the scale-invariant potential and to derive a complementary upper bound for the same quantity. The existence of global solutions when the power nonlinearity is equal to the polynomial root is still an open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21769 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up for a Semilinear Tricomi-type Equation with Scale-Invariant Mass in the Oscillatory Regime Marcon, Diego Nascimento, Wanderley Santos, Matheus Analysis of PDEs We investigate the finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential and power nonlinearities in the oscillatory regime. For smooth, compactly supported, nonnegative initial data, we prove nonexistence of global-in-time solutions when the power nonlinearity lies below the positive root of an explicit Strauss-type polynomial naturally associated with the equation. The proof combines two main ingredients. The first is the construction of a positive adjoint temporal profile, which yields a weighted monotonicity formula and, consequently, a quantitative lower bound for the nonlinear term. The second is a phase-localized test function argument on logarithmic time shells, fitted to capture the oscillatory effects induced by the scale-invariant potential and to derive a complementary upper bound for the same quantity. The existence of global solutions when the power nonlinearity is equal to the polynomial root is still an open problem. |
| title | Blow-up for a Semilinear Tricomi-type Equation with Scale-Invariant Mass in the Oscillatory Regime |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.21769 |