Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem

We consider the global existence of classical solutions and blowup phenomena for a spatially one‐dimensional radiation hydrodynamics model problem, which consists of a scalar Burgers‐type equation coupled with a nonlocal advection‐reaction equation for radiation intensity. The model can be seen as an extension of the well‐known Hamer model that includes additionally the effects of scattering. It is well‐known that the initial value problem for Burgers' equation cannot be solved classically as soon as the derivative of the initial datum is negative somewhere. For our model problem, there is a critical negative number such that if the spatial derivative of the initial function is larger than this number, the associated initial‐value problem admits a global classical solution. However, when the spatial derivative of the initial data is below another negative threshold number, the initial value problem can also not be solved classically. Moreover, when there does not exist a global classical solution, it is shown that the first spatial derivative of solution must blow up in finite time. The results of the paper generalize the findings of Kawashima and Nishibata for the Hamer model. Copyright © 2012 John Wiley & Sons, Ltd.