Blow-Up at Space Infinity for Nonlinear Heat Equations

On Instant Blow-up for Semilinear Heat Equations with Growing Initial Data

“…We consider the solution w of the initial value problem (1) with an initial data w 0 having a direction ψ ∈ S n−1 means convergence in [3]. One of equivalent definition reads: there exists a positive constant M such that 0 ≤ w 0 ≤ M and inf x∈B(xm,rm)…”

“…It turns out that this condition is equivalent to say that the solution has a least blow-up time (see [15], [3] (v(t) − w(x, t)) = 0,…”

“…Moreover, blow-up occurs only at the space infinity. This implies that the profile function w(x, T * ) at at t = T * is continuous (in fact C 2 ) and positive by a standard linear regularity theory [3] (which has an elliptic counterpart [6]). As in [4], [5] lim |x|→∞ w(x, T * ) = ∞.…”

“…This is clearly related to the notion of 'complete blow-up' ( [16], [12]), but it is not the same. If w 0 has a direction of mean convergence which is equivalent to say the solution has a least blow-up time, (i.e., the blow-up time T * agrees with the blow-up time of a spatially constant solution with an initial data sup w 0 ), then the blow-up profile w(x, T * ) has a blow-up direction (see [3]). As remarked in Appendix the w(x, T * ) fulfills (3).…”

“…For the space infinity blow-up the reader is refereed to papers [10], [7] [4], [5], [15] [17], [16], [14] and a review paper [3]. There is a nice book [12] for overview of blow up problems.…”