JournalsaihpcVol. 33, No. 6pp. 1473–1495

Dispersion for 1-D Schrödinger and wave equations with BV coefficients

  • Enrique Zuazua

    BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain, Ikerbasque, Basque Foundation for Science Maria Diaz de Haro 3, 48013, Bilbao, Basque Country, Spain

  • Constantin N. Beli

    Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania

  • Liviu I. Ignat

    Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Street, 010014 Bucharest, Romania
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Abstract

In this paper we analyze the dispersion for one dimensional wave and Schrödinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrödinger equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrödinger equation.

Cite this article

Enrique Zuazua, Constantin N. Beli, Liviu I. Ignat, Dispersion for 1-D Schrödinger and wave equations with BV coefficients. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 6, pp. 1473–1495

DOI 10.1016/J.ANIHPC.2015.06.002