Oscillation of a class of functional parabolic differential equations
The authors of this interesting paper investigate a class of nonlinear parabolic equations with distributed deviating arguments, \[ u_t= a(t)\Delta u+ \sum^n_{i=1} a_i(t)\Delta u(x,\tau_i(t))- p(x,t)u- \int^b_a q(x,t,\xi) f(u[x, g(t,\xi)]) d\sigma(\xi)+ h(x, t), \] \((x,t)\in \Omega\times \mathbb{R}_+\equiv G\), the integral is a Stieltjes integral. Given one of the boundary conditions \(u= \varphi(x,t)\), \(\partial u/\partial n= \psi(x,t)\), \(\partial u/\partial n+\nu(x,t)u= 0\) on \(\partial\Omega\times \mathbb{R}_+\). Here \(\Delta\) is the Laplacian operator in \(\mathbb{R}^n\), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with piecewise smooth boundary \(\partial\Omega\). The functions \(a(t)\), \(a_i(t)\), \(\tau_i(t)\), \(p\), \(q\), \(g\), \(f\) are continuous with respect to their arguments. Interesting oscillatory criteria for the considered problem under each one of the three boundary conditions are established.
nonlinear parabolic equations with distributed deviating arguments
scientific article; zbMATH DE number 1528557