Bernstein functions and parabolic equations in \(\text{BUC}(\mathbb{R}^n,\mathbb{R})\) (Q1348170)

The authors show an application of the known quasimonotonicity method making use of the Bernstein functions to some parabolic differential-functional equations in BUC(\(\mathbb{R}^{ n},\mathbb{R} \)) having the form \[ u_t=\sum_{j,k=1}^na_{j,k}(t)D_jD_ku+\sum_{j=1}^nb_{j}(t)D_ju+\sum_{j=1}^mc_{j}(t)S_{g_j(t)}u \] under initial condition \(u(0)=u_0\). Here \(a_{j,k}(t)\), \(b_{j}(t)\), \(c_{j}(t)\), \(g_j(t)\) are continuous, \(a_{j,k}(t)\) - positive semidefinite, BUC(\(\mathbb{R}^{n},\mathbb{R} \)) is the Banach space of all bounded, uniformly continuous functions on \(\mathbb{R}^{ n}\) endowed with the supremum norm \(\|\cdot \|_{\infty }\). A new proof of a known result on solvability and asymptotic behaviour of the solutions to the considered problem is discussed. Actually, the result is that the solution of the problem under consideration satisfies the inequality \[ \|u(t)\|_{\infty }\leq \exp \int_{t_0}^tc(s) ds\|u_0\|_{\infty } \quad (t\in [ t_0,T)), \] where \(c(t)=\sum_{j=1}^mc_j(t)\).