Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation
(EN)
Tzanetis, DE
(EN)
The initial-boundary value problem for the nonlinear heat equation u(t)=Delta u+lambda f(u) might possibly have global classical unbounded solutions, u*=u(x,t;u(0)*), for some ''critical'' initial data u(0)*. The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;lambda) for some values of lambda. We find, for radial symmetric solutions, that u*(r,t)-->w(r) for any 0<r less than or equal to 1 but supu*(.,t)=u*(0,t)-->infinity, as t-->infinity. Furthermore, if (u) over cap(0)>u(0)*, where u(0)* is some such critical initial data, then (u) over cap=u(x,t;(u) over cap(0)*) blows up in finite time provided that f grows sufficiently fast.
(EN)
