Non autonomous parabolic problems with unbounded coefficients in unbounded domains. (English) Zbl 1334.35073

The authors prove existence and uniqueness results for the nonautonomous Cauchy problem \[ \begin{cases} D_tu(t,x)=(\mathcal{A}u)(t,x) & t\in (s,+\infty), \;x\in \Omega,\\ (\mathcal{B}u)(t,x)=0 & t\in(s,+\infty),\;x\in\partial\Omega,\\ u(s,x)=f(x) & x\in \Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^d\) is unbounded domain with \(C^{2,\alpha}\)-smooth boundary, \(f\) is bounded and continuous function in \(\Omega,\) \(\mathcal{A}\) is a second-order elliptic operator with smooth and unbounded coefficients depending on \((t,x),\) while \(\mathcal{B}\) is zero or first-order non-tangential boundary operator with smooth coefficients that corresponds respectively to the Dirichlet or to a non-degenerate oblique derivative condition.


35K10 Second-order parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
Full Text: arXiv Euclid