Well-Posedness of the Cauchy Problem for Some Evolution Equations

Degenerate Parabolic Differential Equations

“…Here we consider the case where bj(x 9 t) and c(x, t) are complex-valued functions. Put iy = 6j 1) + J6J 2) , where 6J 1) and 6J 2) are real-valued. We assume the following condition: for any (x 9 t)^R n x\J), oo) where £ is an arbitrary small positive constant and C € is a constant dependent on s. The above condition (3.13) is, in some sense, close to being necessary.…”

“…bounded in L 2 (G). Then a weak limit of them, as £-»0 9 gives the desired weak solution u(x, t)^L 2 (G).…”

“…!)-(!. 2) for any initial value u Q (x)^L 2 It seems to us that our method is more natural than the one relying on elliptic regularization and will be useful to other problems. ** 5 §2 0 Energy Inequality Let a jh (x, *)e<^°0# 2 ); b;(x, t)s=£?…”

“…Then a weak limit of them, as £-»0 9 gives the desired weak solution u(x, t)^L 2 (G). The uniqueness of the solution is proved.…”