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The focus of this article is on the
geometric mechanism for the blow-up of solutions to the initial value problem
for scalar conservation laws. We prove that the sufficient and
necessary condition of blow-up is the formation of characteristics envelope.
Whether the solution blows up or not relates to the topology structure of a set
dominated by initial data. At last we take Burger’s equation as an example to
verify our main theorem.
In this paper we focus on the initial value problem of a
hyperbolic-elliptic coupled system in multi-dimensional space of a radiating
gas. By using the method of Green function combined with Fourier analysis, we
obtain the pointwise decay estimates of solutions to the problem.