On solvability of boundary value problems for quasilinear mixed type operator-differential equations (Q2713894)

The article is devoted to the study of boundary value problems for the equation NEWLINE\[NEWLINE B(t)u_t - G(t,u)=f(t), \quad t\in (0,T), \tag{1} NEWLINE\]NEWLINE where \(B(t):E\to E\) is a family of linear operators in a separable Hilbert space \(E\) and \(G(t,\cdot):E\to E\) is a family of nonlinear operators. The operators \(B(t)\) are assumed to be symmetric and the operators \(B(0)\) and \(B(T)\) selfadjoint. Moreover, there exist a Hilbert space \(H_1\) and a Banach space \(V_1\), \(H_1\cap V_1\) is densely embedded into \(E\), such that \(B(t)\in W_\infty^1(0,T;L(H_1\cap V_1,H_1'+V_1'))\) and the operators \(Su=-G(t,u)-\frac{1}{2}B_t(t): H_1\cap V_1\to H_1'+V_1'\) are coercive and monotone almost everywhere on \((0,T)\). The equation (1) is furnished with the boundary conditions NEWLINE\[NEWLINE E^+u(0)=0,\quad \lim_{t\to \infty} u(t)=0, \quad \text{if }T=\infty \qquad \text{and}\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEE^+u(0)=h_{11}E^-(0)u(0)+h_{12}E^+(T)u(T),\;E^-u(T)=h_{21}E^-(0)u(0)+h_{22}E^+(T)u(T),\tag{3}NEWLINE\]NEWLINE if \(T<\infty\), where \(h_{i,j}\) are some linear operators and \(E^{\pm}(0), E^{\pm}(T)\) are the spectral projections of \(B(0)\) and \(B(T)\) corresponding to the positive and negative parts of the spectrum, respectively. Note that the operators \(B(t)\) may have nontrivial kernels. Equations (1) are not Sobolev-type equations and are usually called generalized kinetic equations. NEWLINENEWLINENEWLINEUnder some natural additional conditions, it is demonstrated that the problem (1), (2) ((1), (3)) has a solution \(u(t)\) in the following class: \(u(t)\in L_2(0,T;H_1)\cap L_p(0,T;V_1)\) and \(\frac{d}{dt}B(t)u(t)\in L_2(0,T;H_1')+ L_q(0,T;V_1')\), with \(p\geq 2\) and \(1/p+1/q=1\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].