Regular generalized solutions to semilinear wave equations
- PDF / 334,591 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 45 Downloads / 230 Views
Regular generalized solutions to semilinear wave equations Hideo Deguchi1 · Michael Oberguggenberger2 Received: 19 September 2019 / Accepted: 8 October 2020 © The Author(s) 2020
Abstract The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error. Keywords Semilinear wave equations · Small nonlinearities · Existence of generalized solutions · Algebras of generalized functions Mathematics Subject Classification 35D05 · 35D10 · 46F30 · 35L71
1 Introduction This paper addresses existence and regularity of solutions to semilinear wave equations with a small nonlinearity. The equations are of the form ∂t2 u − Δu = h(ε) f (u), t ∈ [0, T ], x ∈ Rd , u|t=0 = u 0 , ∂t u|t=0 = u 1 , x ∈ Rd
(1.1)
Communicated by Adrian Constantin.
B
Michael Oberguggenberger [email protected] Hideo Deguchi [email protected]
1
Department of Mathematics, University of Toyama, Gofuku 3190, Toyama 930-8555, Japan
2
Arbeitsbereich für Technische Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria
123
H. Deguchi, M. Oberguggenberger
where Δ denotes the Laplacian, h(ε) is a net of positive real numbers tending to zero as ε → 0, f is smooth with f (0) = 0. The initial data u 0 and u 1 are generalized functions of compact support, and the space dimension is d = 1, 2, 3. Approximating the initial data by nets of smooth functions (u 0ε , u 1ε )ε∈(0,1] , the goal of the paper is to establish the existence of a net of smooth solutions (u ε )ε∈(0,1] up to an asymptotic error term of O(ε∞ ). The paper is formulated in the framework of Colombeau generalized functions. We present a new existence result of a global generalized solution without growth or sign restrictions on the nonlinearity f , for initial data possessing so-called G 0 -regularity. It is motivated by a result on propagation of singularities in the onedimensional case of the authors [4]. Our main tool will be the Banach fixed point theorem in the so-called sharp topology, a complete ultra-metric topology on the Colombeau algebras. In the classical literature, the semilinear wave Eq. (1.1) has been studied intensively when h(ε) ≡ 1, with small or with large initial data. The existence or nonexistence of a global classical solution is known to depend on the space dimension, the sign and the growth properties of f (u) as |u| → ∞, and the size of the initi
Data Loading...
