Singular perturbation for an abstract non-densely defined Cauchy problem
This work deals with a class of singularly perturbed non-densely defined abstract Cauchy problems of the form: \[ \varepsilon\frac{du_{\varepsilon}(t)}{dt}=Au_{\varepsilon}(t)+F(u_{\varepsilon}(t),Lv_{\varepsilon}(t)), \] \[ \frac{dv_{\varepsilon}(t)}{dt}=Bv_{\varepsilon}(t)+G(u_{\varepsilon}(t),v_{\varepsilon}(t)) \] for \(t\geq 0\) and the initial data \(u_{\varepsilon}(0)=x_{\varepsilon}\in\overline{D(A)}\), \(v_{\varepsilon}(0)=y_{\varepsilon}\in\overline{D(B)}\). Here, \(\varepsilon>0\) is a small parameter and the initial data satisfy \[ \lim_{\varepsilon\to 0^+}x_{\varepsilon}=x_0\in\overline{D(A)}, \lim_{\varepsilon\to 0^+}y_{\varepsilon}=y_0\in\overline{D(B)}. \] \(A:D(A)\subset X\to X, B:D(B)\subset Y\to Y\) are two given Hille-Yosida linear operators acting respectively on the Banach spaces \(X,Y\). \(L\in{\mathcal L}(Y,Z)\) is a bounded linear operator defined on the Banach space \(Y\) taking values in a Banach space \(Z\). \(F(\cdot,\cdot)\) and \(G(\cdot,\cdot)\) are Lipschitz continuous in bounded sets functions. The paper extends the classical Tikhonov theorem for ordinary differential equations to the case of abstract Cauchy problems. It is proved that the solutions rapidly evolve and stay in some neighbourhood of the slow manifold. One obtains that the solutions of the given Cauchy problem converge on each compact time interval, as the singular parameter goes to zero, towards the solutions of the so-called reduced problem. These results are applied to an example of age-structured model as well as to a class of functional differential equations.
scientific article; zbMATH DE number 6822632
scientific article