Weakly coupled systems of quasilinear boundary value problem with applications to semilinear elliptic equations
The authors develop a qualitative theory of diagonal quasilinear hyperbolic systems of the form \(\partial_t u_i + \partial_x f_i(u_i) = g_i(U)\), \(i=1, \dots ,N\), \(U = (u_1, \dots ,u_N)\), with locally Lipschitz continuous functions \(f_1, \dots ,f_N\), \(g_1, \dots , g_N\). They establish local existence and continuation properties for Kruzhkov-type entropy solutions of the Cauchy problem and use the concept of quasi-monotonicity of the mapping \(G=(g_1, \dots ,g_N)\) for proving comparison theorems for entropy sub- and supersolutions. The general results are then applied to concrete problems arising in chromatography, combustion theory and nonlinear optics.
scientific article; zbMATH DE number 1095813