On the construction of compatible data for hyperbolic-parabolic initial-boundary value problems. (English) Zbl 0921.35013

This article is a sequence of articles published by the author on this subject, and some more will be published later. The author considers the system \[ \varepsilon u_{tt}+u_t-\sum^n_{i,j=1} a_{ij}(x,t) \partial_i \partial_j u=f(x,t); \quad u(x,0)= u_0(x),\;u_t(0)= u_1(x),\;u(.,t)_{| \partial \Omega} =0. \] Using some results published in the author’s earlier paper and the results of Tisio Kato, the author cleverly, by recursive techniques, generates a sequence of functions \(\{u_k\}\) on \(\Omega\) as follows; \[ \varepsilon u_{k+2}= (\partial^k_tf)(.,0)-u_{k+1}+ \sum(^{k}_{j}) (\partial^j_tL) (0)u_{k-j}. \] The author proves that these data satisfy the hyperbolic compatibility conditions for an order higher than the earlier results. It is also shown that the same is true for parabolic problems, and one set of data works for both the HCC and PCC.
Reviewer: H.S.Nur (Fresno)


35B25 Singular perturbations in context of PDEs