Propagation of singularities for generalized solutions to nonlinear wave equations
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Journal of Fixed Point Theory and Applications
Propagation of singularities for generalized solutions to nonlinear wave equations Hideo Deguchi
and Michael Oberguggenberger
Abstract. The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions. Mathematics Subject Classification. Primary 35A21, 46F30, 47H10; Secondary 35L71. Keywords. Semilinear wave equations, propagation of singularities, algebras of generalized functions, ultrametric topology, fixed point theorem.
1. Introduction This paper addresses propagation of singularities for solutions to semilinear wave equations with a small nonlinearity. The equations are of the form: ∂t2 u − ∂x2 u = εf (u), t ∈ [0, T ], x ∈ R, u|t=0 = u0 , ∂t u|t=0 = u1 , x ∈ R,
(1)
where ε is a small positive parameter, and f is smooth, polynomially bounded, and f (0) = 0. The initial data u0 and u1 are generalized functions of compact support, with a singularity at the origin. We approximate the initial data by nets of smooth functions (uε0 , uε1 )ε∈(0,1] and measure their regularity in terms of estimates as ε ↓ 0. The main assertion of the paper is that there exists a net of smooth solutions (uε )ε∈(0,1] solving (1) up to an asymptotic error term of O(ε∞ ), which is regular inside (and outside) the one-dimensional light cone; 0123456789().: V,-vol
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H. Deguchi, M. Oberguggenberger
that is, the initial singularity propagates only along the two characteristic lines emanating from the origin. The paper is formulated in the framework of Colombeau generalized functions. This will allow us to use the powerful tools from this theory to combine generalized function data with nonlinearities and to measure regularity. Our main tool will be the Banach fixed point theorem in the so-called sharp topology, a complete ultrametric topology on the Colombeau algebras. More precisely, the fixed point theorem is applied on the subset of solutions which have the desired regularity property. For this purpose, we introduce an ultrametric that captures the refined directional structure of the solution and thereby allows one to transport the regularity of the initial data into the interior of the light cone. To our knowledge, this is the first time in the literature that such fixed-point arguments have been used to establish existence and regularity of solutions to nonlinear wave equations in the Colombeau f
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