Finite time singularities for the multidimensional second sound equation with radial symmetry
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We study the singularity formation of smooth solutions for the radially symmetric degenerate second sound equation, which describes the propagation of wave of temperature in superfluids. It is verified that, even for an arbitrarily small initial energy, its smooth solution may develop cusp-type singularity in the small period. The main difficulty is to prevent the occurrence of degeneracy of wave speed in a small time. Our proof is based on the method of characteristic and the energy estimate.
Abstract
We study the singularity formation of smooth solutions for the radially symmetric degenerate second sound equation, which describes the propagation of wave of temperature in superfluids. It is verified that, even for an arbitrarily small initial energy, its smooth solution may develop cusp-type singularity in the small period. The main difficulty is to prevent the occurrence of degeneracy of wave speed in a small time. Our proof is based on the method of characteristic and the energy estimate.