An example of blowup produced by equal diffusions
The paper proposes a very elegant example of blowing up solutions of a parabolic system with bounded initial data, produced by equal diffusion constants. Precisely, the author considers the system \[ {{\partial u}\over {\partial t}}=\varepsilon {{\partial^2 u}\over {\partial x^2}}+uv(u-v)(u+1)-\delta u,\quad x\in(-1,1), \] \[ {{\partial v}\over {\partial t}}=\varepsilon {{\partial^2 v}\over {\partial x^2}}+vu(v-u)(v+1)-\delta v,\quad x\in(-1,1),\qquad {{\partial u}\over {\partial x}}= {{\partial v}\over {\partial x}}=0, \quad x=\pm 1. \] It is proved that when \(\varepsilon=0\) and \(\delta>0,\) the origin is a global attractor of the positive solutions to the above system, but when \(\varepsilon\) and \(\delta\) are both positive, there is a positive solution with bounded initial data which blows up before any prescribed positive time \(T.\)
blow-up of solutions to parabolic system
scientific article; zbMATH DE number 1288037
scientific article