On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity
Alessandro Palmieri
TL;DR
The paper analyzes the semilinear Euler-Poisson-Darboux-Tricomi equation with time-dependent damping $\mu t^{-1} \partial_t u$ and mass $\nu^2 t^{-2} u$ under power nonlinearity $|u|^p$, with data prescribed at $t=1$. It shows that the subcritical regime $1<p< p_c$, where $p_c=\max\{ p_{\mathrm{Str}}(n+\frac{\mu}{\ell+1},\ell), p_{\mathrm{Fuj}}((\ell+1)n+\frac{\mu-1}{2}-\frac{\sqrt{\delta}}{2}) \}$ and $\delta=(\mu-1)^2-4\nu^2\ge0$, yields finite-time blow-up via a double multiplier method, an adjoint-weighted functional, and an iterative lower-bound framework for the space-average $U(t)=\int_{\mathbb{R}^n}u(t,x)\,dx$. Lifespan estimates are provided, distinguishing the Struts- and Fujita-type thresholds through the function $\theta(n,\ell,\mu,p)$. A separate result addresses the critical Fujita exponent, employing a slicing technique to obtain exponential-type lifespan bounds that depend on $\delta$. The conclusions unify diffusion- and wave-like behaviors in this damped generalized Tricomi setting and extend the understanding of critical exponents for semilinear EPDT equations, with consistency checks in known special cases (e.g., reduced Tricomi or Einstein–de Sitter regimes).
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.