Asymptotics of blowup for a convection-diffusion equation with conservation
The authors consider the following nonlinear initial-boundary value problem \[ u_t= (u_x+ u^q)_x,\text{ for }(x, t)\in (0, 1)\times (0, T),\;u_x+ u^q= 0,\text{ for }x\in \{0, 1\},\;u(x, 0)= u_0(x), \] where \(q> 2\) and \(0< T\leq \infty\). The initial function is such that finite-time blowup occurs at \(t= T\). The limiting behavior of the solution in a vicinity of the blow up point and the asymptotic growth rate are presented. In proving of these results, comparison principles are employed. For the case \(q= 3\), some numerical illustrations of theoretical results are given.
scientific article; zbMATH DE number 883501