This article is cited in 15 papers

Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

M. Kirane^a, N.-e. Tatar<sup>b

a</sup> Université de La Rochelle
^b King Fahd University of Petroleum and Minerals

Abstract: We consider the Laplace equation in $\mathbb R^{d-1}\times\mathbb R^+\times(0,+\infty)$ with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.

Keywords: critical exponent, dynamical boundary condition, fractional derivative, Laplace equation.

UDC: 517.957

Received: 16.03.2006

 English version: Siberian Mathematical Journal, 2007, 48:5, 849–856

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