On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary Condition

An elementary model for autoignition of laminar jets

“…Existence of a unique classical solution for problem (2.7)-(2.9) with μ sufficiently small, as well as the fact that the solution of this problem blows up provided μ is large enough, were established in [21] (see also [22] for a more general framework).…”

Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions

“…In any case the relation (2.6) holds. Finally, if /, fu are positive increasing and the integral / is finite then by the result of [5] the solution grows unbounded in a finite time provided that u(tx, x) is sufficiently large for some r, > 0. But this follows immediately from the relation (2.6).…”

“…In an earlier paper [5] Walter established the existence and nonexistence of global solutions for the parabolic system u,-Lu=u,-i 2 fyC*)«^ + 2 ¿>,(*K -c(x)u\ = 0 \ij-l i-l / (i>0,xGß), (1.1) ou/dp = fix, u) it>0,xG 3ß), (1.2) «(0, *) = «"(je) ix G ß), (1.3) in a framework of differential inequalities, where ß is a bounded domain in R" with boundary 3ß, L is a uniformly parabolic operator in ß, d/dp is the outward normal derivative on 3ß and / is a continuous nonnegative function on 9ß X [0, oo). The main conclusion in [5] states that if/ =fiu) and if there exists t/0 > 0 such that /, /' are both positive and increasing for u > T/n, then global solutions exist when i=r[Av)f'iv)yidn = oo (1.4) and the solution blows-up in finite time for a class of initial functions when 7 < oo. This nonexistence problem due to a positive nonlinear function on the boundary surface has also been discussed by Levine and Payne [2] and by Pao [3] using a different argument.…”

Precise bounds for finite time blowup of solutions to very general one-space-dimensional nonlinear Neumann problems

“…In particular, it is well known, in any space dimension, that if the initial condition f is of one sign, then blowup in finite time is assured (if the function f changes sign, then the solution need not blow up; see for instance [2]). Early results on blowup for the heat equation with nonlinear boundary conditions were obtained in [9] and [13], where the authors demonstrate the inevitability of blowup for certain types of nonlinear boundary conditions and initial data, as well as for variations of the heat equation itself. Considerable work has been done on the problem (1)-(4) to determine, for example, where in Ω the function u will blow up (in general, on some subset of ∂Ω), to provide upper and lower bounds on the time at which blowup will occur, and to provide upper and lower bounds for the solution near blowup.…”