Time-dependent domains for nonlinear evolution operators and partial differential equations

Abstract

This article concerns the nonlinear evolution equation $$displaylines{ frac{du(t)}{dt} in A(t)u(t), quad 0 leq s < t < T, cr u(s) = u_0 }$$ in a real Banach space X, where the nonlinear, time-dependent, and multi-valued operator $ A(t) : D(A(t)) subset X o X$ has a time-dependent domain D(A(t)). It will be shown that, under certain assumptions on A(t), the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type with time-dependent boundary conditions. Those partial differential equations are studied to a large extent.

Keywords

• Dissipative operators
• evolution equations
• parabolic and elliptic equations