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Existence Theorems for Some Non-Linear Equations of Evolution

Published online by Cambridge University Press:  20 November 2018

John C. Clements
John C. Clements*
Affiliation:
University of Toronto, Toronto, Ontario University of Washington, Seattle, Washington
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In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form

(1-1)

where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by

(1-2)

with

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 726 - 745
Copyright
Copyright © Canadian Mathematical Society 1970

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