The Absence of Solutions of Elliptic Systems with Dynamic Boundary Conditions

Abstract

u =0 ; (t;x)2 (0;1)D; ut+ku = h(t;x;u); (t;x 0 )2 (0;1)@D; (P 1) u =0 ; (t;x)2 (0;1)D; utt+ku = f (u); (t;x 0 )2 (0;1)@D; (P 2) ut u = u 1+ ; (t;x)2 (0;1)D; ut+ku = u 1+ ; (t;x 0 )2 (0;1)@D; (P 3) where D is a bounded domain and k and are positive constants. Conditions under which a solution of a second-order degenerate parabolic equation with appropriate restrictions imposed on some nonlinearities blows up in nite time were given in [2]; moreover, criteria for the blow-up of the solution were obtained there on the basis of appropriate ap rioriestimates. A Fujita type result was recently obtained in [3] for the problem u =0 ; (t;x)2 (0;1) ; ut uxN = u q ; (t;x 0 )2 (0;1)@; u (0;x 0 )= ’ (x 0 ) ;x 0 2 @;