Rate of Type II blowup for a semilinear heat equation

Abstract

A solution u of a Cauchy problem for a semilinear heat equationLet ϕ ∞ be the radially symmetric singular steady state. Suppose that u 0 ∈ L ∞ is a radially symmetric function such that u 0 − ϕ ∞ and (u 0 ) t change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with

Blowup rate estimates for the heat equation with a nonlinear gradient source term

“…An important feature of Type II singularity is that the rates are not unique. This can be seen in the construction of solutions with different rates in [19,20,25,13]. We note that (1.8) has been proved by Conner and Grant [5] in one space dimensional case with Ω = (0, 1), under the assumption u 0 > 0 in (0, 1) so that the solution is monotonically increasing in x.…”

“…We call such type of blowup as Type II blowup; see also [25]. Indeed, this type II singularity is also found in the dead-core problem (cf.…”

Improved conditions for single-point blow-up in reaction–diffusion systems

“…[3,5]), or that the solution is monotone in time. Indeed, for large p, even in the particular case of the scalar problem, there exist radial nonincreasing, single-point blow-up solutions of type II (i.e., such that (1.7) fails); see [10,11,13]. As for the case of monotone in time solutions, it seems that the known proofs of (1.7) for systems (see e.g.…”

Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type

“…Such new Type II blow-up patterns for the semilinear heat equation (6.22) were constructed earlier in [37]; see [46] for extra details. We apply this method to the higher-order equation (1.9), which will require completely different spectral theory and related mathematical tools of matching.…”