Blow-up for a Semilinear Tricomi-type Equation with Scale-Invariant Mass in the Oscillatory Regime

TL;DR

This paper studies finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential, identifying conditions under which solutions cannot exist globally in time.

Contribution

It establishes nonexistence of global solutions below a certain critical power, using novel weighted monotonicity and phase-localized test function techniques.

Findings

Solutions blow up in finite time for subcritical nonlinearities.

Constructs a positive adjoint temporal profile for lower bounds.

Develops a phase-localized test function to handle oscillatory effects.

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.