Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

Abstract

AbstractThis is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and different type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.