×

Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time. (English) Zbl 1074.35042

The authors consider the backward parabolic operator \(L := \partial_t + \sum_{j,k=1}^n \partial_{x_j}(a_{jk}(t,x)\partial_{x_k}) + \sum_{j=1}^n b_j(t,x) \partial _{x_j} + c(t,x)\). All the coefficients are defined in \([0,T]\times \mathbb{R}_x^n\), measurable and bounded; the coefficients \(b_j\) and \(c\) are complex valued; \((a_{jk}(t,x))_{jk}\) is a real matrix for all \((t,x) \in [0,T]\times \mathbb{R}_x^n\) and there is \(\lambda_0 \in (0,1]\) such that \(\sum_{j,k=1}^n a_{jk}(t,x) \xi_j\xi_k \geq \lambda_0 | \xi| ^2\) for all \((t,x)\in [0,T]\times \mathbb{R}_x^n\) and \(\xi \in \mathbb{R}_\xi^n\). Given a functional space \({\mathcal H}\) (in which it makes sense to look for the solutions of the equation \(Lu =0\)) one says that the operator \(L\) has the \({\mathcal H}\)-uniqueness property if, whenever \(u \in {\mathcal H}, Lu =0\) in \([0,T]\times \mathbb{R}_x^n\) and \(u(0,x) = 0\) in \(\mathbb{R}_x^n\), then \(u=0\) in \([0,T]\times \mathbb{R}_x^n\). With \({\mathcal H}_1 := H^1([0,T],L^2(\mathbb{R}^n_x))\cap L^2([0,T],H^2(\mathbb{R}^n_x))\) the authors prove the \({\mathcal H}_1\)-uniqueness property for \(L\) when the coefficients \(a_{jk}\) are \(C^2\) in the \(x\) variables and non-Lipschitz-continuous in \(t\). The regularity in \(t\) is given in terms of a modulus of continuity \(\eta\) satisfying the Osgood condition \(\int_0^1 \frac{ds}{\eta(s)} = +\infty\). It is proved that this condition is also necessary.

MSC:

35K15 Initial value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agmon, S.; Nirenberg, L., Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20, 207-229 (1967) · Zbl 0147.34603
[2] Bardos, C.; Tartar, L., Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rat. Mech. Anal., 50, 10-25 (1973) · Zbl 0258.35039
[3] Bony, J.-M., Calcul symbolique et propagations des singularités pour les équations aux dérivées partielles non-linéaires, Ann. Sci. École Norm. Sup., 14, 209-246 (1981) · Zbl 0495.35024
[4] Colombini, F.; De Giorgi, E.; Spagnolo, S., Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6, 511-559 (1979) · Zbl 0417.35049
[5] Colombini, F.; Del Santo, D., Stricly hyperbolic operators and approximate energies, (Proceedings of the ISAAC Conference, Berlin, 2001 (2003), Kluwer Academic: Kluwer Academic Dordrecht), 253-277 · Zbl 1050.35044
[6] Colombini, F.; Del Santo, D., An example of non-uniqueness for a hyperbolic equation with non-Lipschitz-continuous coefficients, J. Math. Kyoto Univ., 42, 517-530 (2002) · Zbl 1148.35334
[7] Colombini, F.; Jannelli, E.; Spagnolo, S., Non-uniqueness in hyperbolic Cauchy problems, Ann. of Math., 126, 495-524 (1987) · Zbl 0649.35051
[8] Colombini, F.; Lerner, N., Hyperbolic operators having non-Lipschitz coefficients, Duke Math. J., 77, 657-698 (1995) · Zbl 0840.35067
[9] Del Santo, D., A remark on non-uniqueness in the Cauchy problem for elliptic operators having non-Lipschitz coefficients, (Hyperbolic Differential Operators and Related Problems. Hyperbolic Differential Operators and Related Problems, Lecture Notes in Pure and Appl. Math., vol. 233 (2003), Dekker: Dekker New York), pp. 317-320 · Zbl 1205.35059
[10] Fleet, T. M., Differential Analysis (1980), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0442.34002
[11] Ghidaglia, J.-M., Some backward uniqueness results, Nonlinear Anal., 10, 777-790 (1986) · Zbl 0622.35029
[12] Lions, J.-L.; Malgrange, B., Sur l’unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scand., 8, 277-286 (1960) · Zbl 0126.12202
[13] Miller, K., Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rat. Mech. Anal., 54, 105-117 (1973) · Zbl 0289.35046
[14] Mizohata, S., Le problème de Cauchy pour le passé pour quelques équations paraboliques, Proc. Japan Acad., 34, 693-696 (1958) · Zbl 0085.08502
[15] Pliś, A., On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Ac. Pol. Sci., 11, 95-100 (1963) · Zbl 0107.07901
[16] Tarama, S., Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients, Publ. Res. Inst. Math. Sci., 33, 167-188 (1997) · Zbl 0882.35034
[17] Tychonoff, A., Théorème d’unicité pour l’équation de la chaleur, Math. Sbornik, 42, 199-215 (1935) · Zbl 0012.35501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.