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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Oscillatory phenomena associated to semilinear wave equations in one spatial dimension


Authors: T. Cazenave and A. Haraux
Journal: Trans. Amer. Math. Soc. 300 (1987), 207-233
MSC: Primary 35L70; Secondary 35B05, 35B15, 35L20
MathSciNet review: 871673
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Abstract: Let $ g$ be a nonincreasing, odd $ {C^1}$ function and $ l > 0$. We establish that for any solution $ u \in C({\mathbf{R}};H_0^1(0,l))$ of the equation $ {u_{tt}} - {u_{xx}} + g(u) = 0$ and any $ {x_0} \in ]0,l[$, the function $ t \mapsto u(t,{x_0})$ satisfies the following alternative:

either $ u(t,{x_0}) = 0,\forall t \in {\mathbf{R}}$,

or $ \forall a \in {\mathbf{R}}$, there exist $ {t_1}$ and $ {t_2}$ in $ [a,a + 2l]$ such that $ u({t_1},{x_0}) > 0$ and $ u({t_2},{x_0}) < 0$.

We study the structure of the set of points satisfying the first possibility. We give analogous results for $ {u_x}$ and for some other homogeneous boundary conditions.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0871673-2
PII: S 0002-9947(1987)0871673-2
Article copyright: © Copyright 1987 American Mathematical Society