Unboundedness of solutions of time-dependent differential systems of parabolic type
Kukio Kobayashi, Norio Yoshida · Institutional Repositories DataBase (IRDB) · 2002
Unboundedness of matrix solutions of time-dependent differential systems of parabolic type is studied. The key tool is to use the Picone-type identity for strongly elliptic systems. The results about oscillations of solutions are also derived. Beginning with the work of McNabb [10], unboundedness of solutions has been investigated by numerous authors. We refer the reader to Dunninger [4], Jaros, Kusano and Yoshida [5, 6] for scalar parabolic equations, and to Chan [1], Chan and Young [2, 3], Kuks [7], Kusano and Narita [8] for parabolic systems. The purpose of this paper is to modify the results of Chan [1], Chan and Young [2] and obtain the results about the oscillations of matrix solutions. We are concerned with the matrix solutions of the time-dependent differential system of parabolic type ∂W ∂t − P [W ] = 0 in Ω ≡ G× (0,∞), (1) where G is a bounded domain in Rn with piecewise smooth boundary ∂G and P [W ] ≡ n ∑ i,j=1 ∂ ∂xi ( Gij(x, t) ∂W ∂xj ) + H(x, t)W. It is assumed that : 2000 Mathematics Subject Classification. 35B05.