Reaction-diffusion equations with nonlinear boundary delay
The following problem (with a boundary condition including a small delay) is considered: \(u_t-\text{Div}(a\nabla u)+ \sum^n_{j=1} B_j(x) {\partial u\over\partial x_j}+ \lambda u=f(u)\), in \(\Omega\times \mathbb{R}^+\), \({\partial u\over \partial n_a}=g(u(t), u(t-r))\) in \(\mathbb{R}^+\), \(u=0\) on \(\partial \Omega\times [-r,0]\); \(\Omega\) is an open bounded domain in \(\mathbb{R}^n\), \(r>0\) is the delay, \(a\in C^1(\overline \Omega)\), \((a(x)>m_0>0\), \(x\in\Omega)\), \(\partial u/ \partial n_a=\langle a\nabla u,\vec n\rangle\) \((n\) is the outward normal), \(\lambda\) is a positive constant, \(B_j(x)\) is continuous in \(\overline \Omega\) \((j=1, \dots,n)\), \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}^2 \to\mathbb{R}\) are smooth functions. Using semigroup theory, global existence and uniqueness of solutions in a convenient fractional power space \((H^s_p(\Omega))\) is shown. It is also proved that all bounded solutions are asymptotic to the set of equilibria as \(t\to\infty\).
scientific article; zbMATH DE number 1333042