| Abstract: |
| This talk focuses on the singularity formation for smooth solutions to the compressible radially symmetric Euler equations with damping. A smooth solution is called a supersonic inward wave if it satisfies $u<-h<0$. For polytropic gases with $\gamma\geq 3$, the damping term introduces additional negative contributions into the evolution equations, which significantly modifies the structure of the corresponding Riccati type inequalities and accelerates the growth of gradients. By applying the characteristic method and constructing suitable invariant domains which different from the case of expanding wave, we prove that smooth supersonic inward solutions with sufficiently strong initial compression must develop singularity in finite time. Furthermore, an explicit upper bound for the lifespan of solutions is derived, which quantitatively characterizes the blowup mechanism of inward wave under damping. This work develops the studies by G. Chen et al. (arXiv: 2511.15180, 2025). |
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