Strong solution for singularly nonautonomous evolution equation with almost sectorial operators
Maykel Belluzi, Tomás Caraballo, Marcelo J. D. Nascimento, Karina Schiabel, Universidad de Sevilla, Spain · Discrete and Continuous Dynamical Systems · 2022
In this paper we consider the singularly nonautonomous evolution problem $ u_t +A(t) u = f(t), \mbox{ } \tau<t<\tau+T; \quad u(\tau) = u_0 \in X, $ associated with a family of uniformly almost sectorial linear operators $ A(t):D\subset X \rightarrow X $, that is, a family for which a sector of the complex plane is contained in the resolvent of $ -A(t) $ and satisfies $ \|(\lambda+A(t))^{-1}\|_{\mathcal{L} (X)} \leq \frac{C}{|\lambda|^{\alpha}} $, for some $ \alpha \in (0, 1) $, uniformly in $ t $. Under a proper condition on the value of $ \alpha $ we prove that the linear process associated to the family $ A(t) $, $ t\in \mathbb{R} $, is strongly differentiable and that the singularly nonautonomous problem has a strong solution. An example of a singularly nonautonomous reaction-diffusion equation in a domain with a handle illustrates the abstracts results obtained.