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Oscillation of nonlinear parabolic equations with functional arguments. (English) Zbl 0614.35048
Some results are established concerning the oscillation of solutions to the nonlinear parabolic equations \[ u_ t=a(t)\Delta u\pm q(x,t)f(u(x,\sigma (t))),\quad (x,t)\in \Omega \times {\mathbb{R}}_+, \] where \(\Delta\) is the Laplacian in \({\mathbb{R}}^ n\) and \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\). The functional argument \(\sigma\) (t) is required to be continuous and satisfy \(\lim_{t\to \infty}\sigma (t)=\infty\). The solution u is oscillatory provided it has a zero in \(\Omega\times [\tau,\infty)\) for every \(\tau >0\). Two types of boundary conditions are imposed, either \(u=0\) or \(\partial u/\partial \nu +\mu u=0\) on \(\partial \Omega \times {\mathbb{R}}_+\). The method of proof is to reduce the problem to a one-dimensional oscillation problem for ordinary differential equations or differential inequalities involving the first eigenfunction of the problem \(\Delta v+\lambda v=0\) in \(\Omega\), \(v=0\) on \(\partial \Omega\). A number of examples are given to illustrate the results.
Reviewer: G.Webb

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs