On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity

TL;DR

This paper investigates the blow-up behavior of solutions to a semilinear generalized Tricomi equation with damping and mass terms that have critical time-dependent decay rates, revealing how these coefficients influence the critical exponent.

Contribution

It derives a blow-up result for the equation with critical decay rates, highlighting the interplay between Fujita-type and Strauss-type exponents influenced by damping and mass coefficients.

Findings

Critical decay rates determine the blow-up threshold.

The critical exponent is affected by the coefficients of damping and mass.

A competition between Fujita-type and Strauss-type exponents is established.

Abstract

In this note, we derive a blow-up result for a semilinear generalized Tricomi equation with damping and mass terms having time-dependent coefficients. We consider these coefficients with critical decay rates. Due to this threshold nature of the time-dependent coefficients (both for the damping and for the mass), the multiplicative constants appearing in these lower-order terms strongly influence the value of the critical exponent, determining a competition between a Fujita-type exponent and a Strauss-type exponent.