Blow-up in a reactive-diffusive medium with a moving heat source
The authors find sufficient conditions for finite time blow-up and global existence of solutions of the Cauchy problem \(u_t-u_{xx}=\delta(x-x_0)F(u(x_0,t))\) in \(\mathbb R\times\mathbb R^+\), \(u(x,0)=\widehat u(x)\geq 0\), where \(\widehat u\in C_0(\mathbb R)\), \(x_0=x_0(t)\) is a prescribed continuous function, \(F\geq 0\) is a smooth increasing convex function and \(\delta\) is the Dirac distribution. It is shown that if the speed \(x_0'(t)\) of the heat source remains sufficiently high then the solution exists globally. On the other hand, if this speed remains sufficiently low and \(\int^\infty(1/F(t)) dt<\infty\) then the solution cannot exist globally. The results are obtained by studying an integral equation satisfied by \(u(x_0(t),t)\). Some special examples are examined.
scientific article; zbMATH DE number 1743199