A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations

Abstract:

We consider the solution operator \[ (Lu)(t) = A\int _0^t {G(t - \tau )} {A^{ - 1}}Bu(\tau )d\tau \] corresponding to the abstract equation $\dot x = Ax + Bu$ on a reflexive Banach space $X$, where the linear operator $A:X \supset \mathcal {D}(A) \to X$ is the infinitesimal generator of a (strongly continuous) group $G(t)$ of bounded operators on $X$, and $B:U \supset \mathcal {D}(B) \to X$ is a generally unbounded linear operator with ${A^{ - 1}}B \in \mathcal {L}(U,X),U$ being another reflexive Banach space (without loss of generality we take $A$ to be boundedly invertible). Let $0 < T < \infty$ be given. We prove the following theorem: if $L$ is continuous ${L^p}(0,T;U) \to {L^p}(0,T;X), 1 < p < \infty$, then in fact $L:$ continuous ${L^p}(0,T;U) \to C[0,T];X$, a lifting regularity theorem in the time variable. Moreover, we show by a parabolic example with nonhomogeneous term in the Dirichlet boundary conditions that the theorem fails to be true, if $G(t)$ is merely a s.c. semigroup even if holomorphic. Applications of the theorem include mixed hyperbolic problems, including second order scalar hyperbolic equations defined on an open bounded domain $\Omega \subset {R^n},\partial \Omega = \Gamma$, with nonhomogeneous term of class ${L^2}(0,T;{L^2}(\Gamma ))$ acting in the Dirichlet or in the Neumann boundary conditions. In the former case, the theorem recovers the authors’ original procedure which yielded optimal regularity results for this dynamics [L-T.2]; in the latter, the theorem improves upon results of Lions-Magenes [L-M.1, vol. II]. Extension to $T = \infty$ is also studied.