Absorption evolution families and exponential stability of non-autonomous diffusion equations (Q1854057)

Let \( \mathcal U = \{ U(t,s)\}_{(t,s)}\), \(t,s \in I\) (\(I\) is an interval) be a strongly continuous evolution family on \(X= L^p(\Omega)\), \(1 \leq p < \infty\), \(V \in L^\infty (I \times \Omega) \), then the existence of a strongly continuous evolution family \(\mathcal U_V = \{U_V(t,s)\}_{(t,s)}\) fulfilling \[ U_V(t,s)x = U(t,s)x- \int _s^t U(t,\tau)V(\tau)U_V(\tau ,s)x d\tau \] for \(t,s \in I \) is proved. \( \mathcal U_V \) is uniqely determined by this equation. If the condition \( V \in L^\infty (I \times \Omega) \) is substituted by conditions: \(\mathcal U\) is positive, \(V\) nonnegative, then \(\mathcal U_V \) is called absorption evolution family corresponding to \(\mathcal U, V\). The authors give additional assumptions (e.g. \(U(t,s)X \subset D(V(t))\) under which \(\mathcal U_V\) is a strongly continuous family in such cases, too. The last section is devoted to the exponential stability of \(\mathcal U , V\). One of the results can be formulated as follows: If \(\mathcal U\) is a submarkovian, strongly continuous evolution family on \(X= L^1(R^N)\) satisfying a lower Gaussian estimate and \[ 0 \leq V \in \overline {L^\infty (I \times \mathbb{R}^N)}, \quad \int _s^{s+t} \int _{B(\xi ,r)} V(\tau ,\eta) d\eta d\tau \geq c >0, \] for some \(t>0\) and all \(s \in I\) and with \(\xi \in \mathbb{R}^N,\) then \(\mathcal U_V\) is exponentially stable where \(c,r>0\) are constants and \(B(\xi ,r)\) is a ball with centre \(\xi \) and radius \(r\). The closure is considered in the space \(L_{\text{loc},u}^1\). These results can be applied to the equation \[ \frac {du}{dt}=\sum _{k,l=1}^N D_k a_{k,l}(t) D_lu(t))-V(t) u(t) \] where \(a_{k,l} \in L^\infty (\mathbb{R}_+ \times \mathbb{R}^N,\mathbb{R}) \) and the nonnegative function \(V\) fulfils some regularity conditions.