A norm estimate for evolution operators of equations with the Lipschitz property in a Banach space (Q2418839)

Let \(X\) be a Banach space and \(A_0\) be a generator of a \(C_0\)-semigroup. Let \(B(t)\) (\(t\geq0\)) be an operator valued function with the Lipschitz property \[ \|B(t)-B(\tau)\|\leq q_0|t-\tau|\quad (t,\tau\geq0). \] Let \(U(t,s)\) be an evolution family for the Cauchy problem of the equation \[ \frac{du(t)}{dt}=(A_0+B(t))u(t)\quad (t\geq0). \] Assume that for each positive \(\tau\) the operator \(A(\tau)=A_0+B(\tau)\) generates a semigroup such that \[ \|\exp(A(\tau)t)\|\leq a_\nu e^{\nu t}\quad (t\geq0) \] with some positive \(\nu\) and \(a_\nu\). It is proved that for all \(t\geq t_0\geq0\) one has \[ \|U(t,t_0)\|\leq a_\nu e^{\nu(t-t_0)} \cosh(\sqrt{a_\nu q_0}(t-t_0))\leq a_\nu e^{(\nu+\sqrt{a_\nu q_0})(t-t_0)}. \] An application to the equation \[ \frac{\partial u(t,x)}{\partial t}= \frac{\partial}{\partial x}c(x) \frac{\partial u(t,x)}{\partial x}+ b(t)u(t,x)\quad (0\leq x\leq 1) \] with boundary conditions \[ u(t,0)=u(t,1)=0 \] is given.