The book under review is concerned with the existence, uniqueness, regularity properties and asymptotic behaviour of the strong and weak solutions to some classes of nonlinear initial-boundary value problems. For this aim, the author uses several results from the theory of monotone operators and nonlinear evolution equations of monotone type in Hilbert spaces.
In Chapter 1 the author studies the first-order nonlinear hyperbolic system on the positive semi-axis of the spatial variable
\[ \begin{cases} u_t(t,x)+v_x(t,x)+\alpha(x,u)=f(t,x),\\ v_t(t,x)+u_x(t,x)+\beta(x,v)=g(t,x),\end{cases} \qquad t>0,\quad x>0,\quad \text{in}\;\mathbb R^n, \]
with the boundary condition
\[ \binom{u(t,0)}{S(w'(t))}\in -G\binom{v(t,0)}{w(t)}+ B(t),\qquad t>0,\quad \text{in}\;\mathbb R^{n+m} \]
and the initial data \(u(0,x)=u_0(x),\quad v(0,x)=v_0(x),\quad x>0,\quad w(0)=w_0\), in the general case \(B(t)\not\equiv \text{const.}\) Some considerations for the above system in the case \(x\in \mathbb R\) are also addressed.
In Chapter 2 the author investigates the second-order nonlinear hyperbolic system
\[ \begin{cases} u_t(t,x)+v_{xx}(t,x)+\alpha(x,u)=f(t,x),\\v_t(t,x)-u_{xx}(t,x)+\beta(x,v)= g(t,x),\end{cases} \qquad t>0,\quad x>0,\quad \text{in}\;\mathbb R^n, \]
with the boundary condition
\[ \binom{\text{col}(-u_x(t,0),u(t,0))}{S(w'(t))}\in -G\binom{\text{col}(v(t,0),v_x(t,0))}{w(t)}+B(t),\qquad t>0,\quad \text{in}\;\mathbb R^{2n+m} \]
and the initial conditions \(u(0,x)=u_0(x),\quad v(0,x)=v_0(x),\quad x>0\), \(w(0)=w_0\), for a general vector \(B(t)\). The case \(x\in \mathbb R\) for this system, without boundary condition, is also presented.
For the above problems the existence of periodic solutions is also investigated. These problems have applications in integrated circuits modelling, hydrodynamics, gas dynamics and elastic beam theory. For applications and examples, the author refers the reader to her monograph [R. Luca-Tudorache, Boundary value problems for nonlinear hyperbolic systems and applications. Casa de Editura Venus, Iaşi (2003; Zbl 1063.35003)].
Using some theorems from V. Barbu [J. Fac. Sci., Univ. Tokyo 19, 295–319 (1972; Zbl 0256.47052)], in Chapter 3 the author investigates the nonlinear differential systems of the form
\[ \begin{cases} u_{tt}(t,x)=A(v)+\alpha(x,u)+f(t,x),\\v_{tt}(t,x)=B(u)+\beta(x,v)+g(t,x), \end{cases}\qquad t>0,\quad \text{in}\;\mathbb R^n, \]
for \(x\in (0,1)\) or \(x\in (0,\infty)\) or \(x\in \mathbb R\), where \(A(v)=v_x\) and \(B(u)=u_x\), or \(A(v)=v_{xx}\) and \(B(u)=-u_{xx}\), or \(A(v)=\sum_{k=0}^na_k(x)\frac{\partial^k v}{\partial x^k}\) and \(B(v)=-\sum_{k=0}^n(-1)^k\frac{\partial^k}{\partial x^k}[a_k(x)u]\), with different boundary conditions and/or boundedness conditions in the space \(L^2\).
In Chapter 4 she studies three Volterra integro-differential boundary value problems, by applying some results from I. I. Vrabie [Compactness methods for nonlinear evolutions. Harlow: Longman (1987; Zbl 0721.47050)]. Here she investigates local and global existence of the strong and weak solutions, and the uniqueness of solutions.
In Chapter 5 there are studied the nonlinear differential system with first-order differences in a Hilbert space \(H\),
\[ \begin{cases} u_j'(t)+\frac{v_j(t)-v_{j-1}(t)}{h_j}+A(u_j(t))\ni f_j(t),\\ v_j'(t)+\frac{u_{j+1}(t)-u_j(t)}{h_j}+B(v_j(t))\ni g_j(t), \end{cases}\qquad j=1,\dots,\quad t>0, \]
with the extremal conditions \(v_0(t)\in -\alpha(u_1(t))\), \(u_{N+1}(t)\in \beta(v_N(t)),\quad t>0\), and the initial data \(u_j(0)=u_{j0}\), \(v_j(0)=v_{j0}\), \(j=1,\dots,N\), and the nonlinear infinite differential system
\[ \begin{cases} u_n'(t)+\frac{v_n(t)-v_{n-1}(t)}{h}+A(u_n(t))\ni f_n(t),\\v_n'(t)+\frac{u_{n+1}(t)-u_n(t)}{h}+B(v_n(t))\ni g_n(t), \end{cases}\quad n=1,2,\dots,N, \quad t>0, \]
with the extremal condition \(v_0(t)\in -\alpha(u_1(t))\), \(t>0\) and the initial data \(u_n(0)=u_{n0}\), \(v_n(0)=v_{n0}\), \(n=1,2,\dots\) For \(H=\mathbb R\) these problems are discrete versions with respect to \(x\) (\(x\in (0,1)\) or \(x\in (0,\infty)\)) of some nonlinear boundary value problems ((S)\(_0\)+(BC)\(_0\)+(IC)\(_0\) and (S)\(_0'\)+(BC)\(_0'\)+(IC)\(_0'\) in the book). We mention that for the proof of Theorem 5.1.2, the author presents a generalization of G. J. Minty’s theorem to product spaces with weights for the characterization of the maximal monotonicity of the monotone operators.