Phase space for Sobolev type equations with \(s\)-monotone and strongly coercive operators (Q1902850)

Let \(X= (X;\langle\cdot, \cdot\rangle)\) be a real separable Hilbert space identified with its adjoint and \(Y_L\), \(Y_M\) be reflexive Banach spaces. Let the following dense and continuous embeddings hold \[ Y_M\hookrightarrow Y_L \underline{\hookrightarrow} X \underline{\hookrightarrow} Y^*_L\hookrightarrow Y^*_M. \] The authors consider the Cauchy problem \[ (1)\qquad Lu'+ M(u)= 0,\qquad (2)\qquad u(0)= u_0, \] where \(L: Y_L\to Y^*_L\) is a Fredholm, selfadjoint, nonnegative definite and continuous linear operator and \(M\in C^k(Y_M, Y^*_M)\), \(k\geq 1\), is \(s\)-monotone (i.e., \(\langle M_u'v, v\rangle> 0\) for any \(u,v\in Y_M\setminus\{0\}\) and \(M_u'\) denotes the Fréchet derivative of \(M\) at the point \(u\)) and strongly coercive (i.e., \(\lim_{\| v\|\to \infty} {\langle M(u+ v),v\rangle\over\| v\|}= \infty\) for any \(u\in Y_M\)). A set \({\mathcal A}\subset Y_M\) is called a phase space of (1) if (i) any solution \(u(t)\) of (1) lies in \({\mathcal A}\), i.e., \(u(t)\in{\mathcal A}\) for \(t\in[0,\infty)\) and (ii) for any \(u_0\) from a certain dense subset \({\mathcal A}^0\) of \({\mathcal A}\) there exists a unique solution of (1), (2). In the paper, a phase space of (1) is obtained. Some applications of the general theory for the existence and uniqueness results to the generalized Boussinesq filtration equation and an equation which simulates certain diffusion processes are given.