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Non-uniqueness and Uniqueness in the Cauchy Problem of Elliptic and Backward-Parabolic Equations

  • Daniele Del Santo
  • Christian P. Jäh
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

Abstract

In this paper we consider the non-uniqueness and the uniqueness property for the solutions to the Cauchy problem for the operators
$$\mathcal{E} u = \partial_t^2 u + \sum _{k,l=1}^n \partial_{x_k} \bigl(a_{kl}(t,x)\partial_{x_l}u\bigr) + \beta(t,x) \partial_t u + \sum_{m=1}^n b_m(t,x)\partial_{x_m}u + c(t,x)u $$
and
$$\mathcal{P} u = \partial_t u + \sum_{k,l=1}^n \partial_{x_k}\bigl(a_{kl}(t,x)\partial_{x_l}u\bigr) + \sum_{m=1}^n b_m(t,x) \partial_{x_m}u + c(t,x)u, $$
where \(\sum_{k,l=1}^{n} a_{kl}(t,x)\xi_{k} \xi_{l} |\xi|^{-2}\geq a_{0} >0\). We study non-uniqueness and uniqueness in dependence of global and local regularity properties of the coefficients of the principal part. The global regularity will be ruled by the modulus of continuity of a kl on [0,T] while the local regularity will concern a bound on | t a kl (t,x)| on every interval [ε,T]⊆(0,T]. By suitable counterexamples we show that our conditions seem to be sharp in many cases and we compare our statements with known results in the theory of hyperbolic Cauchy problems. We make also some remarks on continuous dependence for \(\mathcal{P}\).

Keywords

Uniqueness Non-uniqueness Cauchy problem Elliptic equations Backward Cauchy problem Parabolic equations Modulus of continuity Osgood condition 

Mathematics Subject Classification

35Bxx 35J15 35K10 35B35 35B60 

Notes

Acknowledgements

Since most of the results are content of his Master thesis the second author would like to thank his supervisor Professor Michael Reissig for turning his attention to the subject of this paper and many helpful and clarifying discussions. He also thanks the Department of Mathematics and Geosciences of the University of Trieste for its warm and inspiring hospitality during several stays in Trieste.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica e GeoscienzeUniversità di TriesteTriesteItaly
  2. 2.Institut für Angewandte AnalysisTU Bergakademie FreibergFreibergGermany

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