An existence result for a nonlinear hyperbolic system.(English)Zbl 0814.35071

The paper gives results for the existence and uniqueness of strong and weak solutions for the nonlinear hyperbolic system \begin{aligned} \ell_ k(t,x) {\partial i_ k \over \partial t} + {\partial v_ k \over \partial x} + \gamma_ k (t,x) \alpha_ k (x,i_ k)&= f_ k(t,x) \\ c_ k(t,x) {\partial v_ k \over \partial t} + {\partial i_ k \over \partial x} + \delta_ k (t,x) \beta_ k (x,v_ k) &= g_ k(t,x) \end{aligned} \tag{1} for $$0 < x < 1$$, $$0 < t < T$$ and $$k = 1, \dots, n$$ with boundary conditions $\begin{split} \left[ \begin{matrix} \text{col} \bigl( i_ 1(t,0), - i_ 1(t,1), \dots, i_ n(t,0), - i_ n(t,1) \bigr) \\ \text{col} \bigl( h_ 1(t) {dw_ 1 \over dt} (t), \dots, h_ m(t) {dw_ m \over dt} (t) \bigr) \end{matrix} \right] \in \\ \in B \left[ \begin{matrix} \text{col} \bigl( v_ 1 (t,0), v_ 1(t,1), \dots, v_ n(t,0), v_ n(t,1) \bigr) \\ \text{col} \bigl( w_ 1(t), \dots, w_ m(t) \bigr) \end{matrix} \right] \end{split} \tag{2}$ for $$0 < t < T$$ and initial conditions $$i_ k (0,x) = i_ k^ 0(x)$$, $$v_ k (0,x) = v_ k^ 0(x)$$, $$w_ j(0) = w^ 0_ j$$ for $$0 < x < 1$$. System (1) generalizes the equations of electrical multiports connected to transmission lines. In (2), $$B$$ is a maximal monotone mapping (possibly multivalued).

MSC:

 35L50 Initial-boundary value problems for first-order hyperbolic systems 47H05 Monotone operators and generalizations 47H04 Set-valued operators 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35R70 PDEs with multivalued right-hand sides