Transient behavior of solutions to a class of nonlinear boundary value problems

Abstract

Abstract. In this paper we consider the asymptotic behavior in time of solutions to the heat equation with nonlinear Neumann boundary conditions of the form ∂u/∂n = F (u), where F is a function that grows superlinearly. Solutions frequently exist for only a finite time before "blowing up." In particular, it is well known that solutions with initial data of one sign must blow up in finite time, but the situation for sign-changing initial data is less well understood. We examine in detail conditions under which solutions with sign-changing initial data (and certain symmetries) must blow up, and also conditions under which solutions actually decay to zero. We carry out this analysis in one space dimension for a rather general F , while in two space dimensions we confine our analysis to the unit disk and F of a special form.

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

“…As noted in the introduction, if the initial condition f changes sign, then blowup need not occur. In [2] we show that sign-changing solutions with certain symmetries may in fact decay to zero if f is "small enough", while other solutions with the same symmetries must blow up in finite time.…”

“…The solution u(x, t) will frequently "blow up", that is, become unbounded, in finite time. 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.…”

Existence of weak positive solutions to a nonlinear PDE system around a triple phase boundary, coupling domain and boundary variables

“…The equation and the boundary conditions of this model are the same as the ones appearing in electrolyte region in PEM fuel cells. In [6] (and similarly in [7,8] for the associated 1D and 2D transient models) it is proved (and conjectured for the general case) that in the case of a circular domain there are an infinite number of solutions (of the potential) which blowup as a certain parameter tends to zero. Nonlinear elliptic and parabolic equations with singular Dirichlet or Neumann boundary conditions has been considered in literature, see for example [4,9,17] and the references within.…”

“…Assume the conditions (4)-(8) hold. There exists w m > u m such that if (u, w) ∈ B is a fixed point of T thenw C 0 (Ω 2 ) w m .…”