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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

##### MSC:
 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