Abstract: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.
| Subjects: | Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2605.21769 [math.AP] |
| (or arXiv:2605.21769v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.21769 arXiv-issued DOI via DataCite |
From: Matheus Correia Dos Santos [view email]
[v1]
Wed, 20 May 2026 22:03:36 UTC (39 KB)
[v2]
Fri, 22 May 2026 11:44:40 UTC (39 KB)
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