# zbMATH — the first resource for mathematics

Estimate for the solution to the Cauchy problem for an ultrahyperbolic inequality. (English. Russian original) Zbl 1146.35099
Dokl. Math. 74, No. 2, 751-754 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 410, No. 6, 737-740 (2006).
The author considers the differential inequality $|Lu|^2\leq C(F^2+ u^2+ |\nabla_x u|^2),$ where $$x\in\Omega_1\times \Omega_2\equiv\Omega$$ with the unknown function $$u(x)$$. Here $$x= (x_1,\dots, x_n)$$, and $$Lu= \sum^n_{i,j=1} (a_{ij}(x) u_{x_i})_{x_j}$$, $$F= F(x)$$ and the coefficients of the operator $$L$$ satisfy the certain conditions. The author proves Carleman-type inequalities for $$L$$ and additionally establishes a Hölder estimate for the corresponding Cauchy problem for $$L$$.

##### MSC:
 35R45 Partial differential inequalities and systems of partial differential inequalities 35B45 A priori estimates in context of PDEs
Full Text:
##### References:
 [1] S. P. Shishatskii, Dokl. Akad. Nauk SSSR 213, 49–50 (1973). [2] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics (Nauka, Moscow, 1980; Am. Math. Soc., Providence, R.I., 1986). [3] D. P. Kostomarov, Dokl. Math. 67, 377–381 (2003) [Dokl. Akad. Nauk 390, 443–447 (2003)]. [4] D. P. Kostomarov, Dokl. Math. 69, 425–427 (2004) [Dokl. Akad. Nauk 396, 597–600 (2004)]. [5] D. P. Kostomarov, Dokl. Math. 71, 84–88 (2005) [Dokl. Akad. Nauk 400, 449–453 (2005)]. [6] D. P. Kostomarov, Differ. Equations 42, 261–268 (2006) [Differ. Uravn. 42, 245–251 (2006)]. · Zbl 1131.35370 · doi:10.1134/S0012266106020133 [7] V. G. Romanov, Dokl. Math. 72, 964–968 (2005) [Dokl. Akad. Nauk 405, 730–734 (2005)]. [8] V. G. Romanov, Sib. Mat. Zh. 47(1), 169–187 (2006). · doi:10.1007/s11202-006-0016-7 [9] D. Tataru, Commun. Partial Differ. Equations 20(5/6), 855–884 (1995). · Zbl 0846.35021 · doi:10.1080/03605309508821117 [10] D. Tataru, J. Math. Pure Appl. 75, 367–408 (1996).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.