In the present paper, the authors study the following semilinear Cauchy problem for generalized Tricomi equations: \begin{align*} & u_{tt}- t^{2m}\Delta u=|u_t|^p,\quad (t,x)\in (0,\infty)\times \mathbb{R}^n,\\ & u(0,x)=\varepsilon f(x), \quad u_t(0,x)=\varepsilon g(x),\quad x \in \mathbb{R}^n, \end{align*} where \(n \geq 1\), \(m \geq 0\), \(p>1\) and \(\varepsilon >0\) is sufficiently small. The nonlinearity is of derivative type. This model is for \(m>0\) weakly hyperbolic at \(t=0\) and has for \(t \to \infty\) an increasing time-dependent coefficient. The authors are interested in blow-up results and estimates of the lifespan. It is clear that such results are only of interest if they provide a local (in time) existence result, too.
The authors prove for \(m \in [0,2)\) and a suitable range of exponents \(p\) such a local (in time) existence result. Here they use the scale-invariance structure of the linear part of the generalized Tricomi equation, which allows to apply special function theory. In particular, the weakly hyperbolic theory gives the correct Sobolev regularity for the data. Then a contraction property can be proved for operators appearing in the definition of weak solutions. This procedure leads to lifespan estimates to below. Maybe in the future the authors use decay properties of Sobolev solutions to Cauchy problems for linear generalized Tricomi equations to get on the one hand global (in time) small data existence results and on the other hand better estimates of the life-span to below.
To verify blow-up phenomena and derive lifespan estimates to above the authors use the test function method. Here they use an interesting family of test functions. All the obtained results lead to a conjecture for the critical exponent and optimal lifespan estimates.
It would be nice to have local (in time) existence results for \(m \geq 2\) and \(p>1\), too. Moreover, global (in time) small data existence results and better estimates of the lifespan to below should be established to verify these conjectures.