Universit at Konstanz Weakly Hyberbolic Equations in Domains with Boundaries Weakly Hyperbolic Equations in Domains with Boundaries
نویسندگان
چکیده
We consider weakly hyperbolic equations of the type u tt (t)+a(t)Au(t) = f(t; u(t)), u(0) = u 0 , u t (0) = u 1 ,u(t) 2 D(A), t 2 0; T], for a function u : 0; T] ?! H, T 2 0; 1], H a separable Hilbert space, A being a non-negative, self-adjoint operator with domain D(A). The real function a is assumed to be non-negative, continuous and (piecewise) continuous diierentiable, and the derivative a 0 will have to satisfy an integrability condition, which will admit innnitely many oscillations near the point of degeneration. For given initial data u 0 ; u 1 a global existence theorem in C((0; T]; D(A s)) is proved for the linear problem f f(t). If a 0 does not change sign, the result can be improved, and nally a local (in time) existence theorem can be proved for nonlinearities f essentially satisfying the mapping property f(; D(A s)) D(A s), where s > 0 describes the regularity class. In the applications, A will be a uniformly elliptic operator in a domain , being a bounded domain with smooth boundary in R n , n 2, for second-order operators then describing a weakly hyperbolic wave equation.
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