Diffusion in Poro-Elastic Media

Cauchy Problem, II. Here we consider the case in which the second operator is symmetric. Assume B : D V is a linear monotone operator with domain D V , where V is a Hilbert space, and that A : V V is a continuous linear symmetric monotone operator. Denote Ž .1 2 the space V with the seminorm A , by V . Then the injection a V V is continuous, and we have V V . V is a Hilbert space, for a a a which we have the identity f u f , Au , f V , u V . Ž . ž / a Va Ž . Ž . Assume that Ker B Ker A . Define the operator on the Hilbert space V by a Au B u for some u D. Ž . Ž . Then for any Dom B D we obtain , Au , Bu Bu u Ž . Ž . Ž . Ž . V V a a Ž . from the identity above, so if B is monotone or sectorial , then is Ž . likewise V -accretive or sectorial . a Ž . The equation f in V is equivalent to a u V : Au f , B u , Ž .