Existence and non-existence of global smooth solutions for \(p\)-system with relaxation
The authors consider the Cauchy problem \(v_t-u_x=0\), \(u_t+p(v)_x ={1\over \varepsilon}(f(v)-u)\), with the initial conditions: \((v,u)(x,0)= (v_0(x), u_0(x))\), where \(\varepsilon>0\) is the relaxation time, \(p(v)\) satisfies several stated conditions. The initial conditions yield that the system has the eigenvalues \(\lambda(v)= -\sqrt{-p'(v)}\), \(\mu(v)= \sqrt{-q'(v)}\), and with the substitutions \(r=u+\Phi(v)\), \(s=u-\Phi(v)\), where \(\Phi(v)= \int_v^{v_0(x)} \mu(\tau) d\tau\), the above system is transformed to this diagonal form \(r_t+\lambda (v)r_x= -{1\over 2\varepsilon} (r+s-F(r-s))\), \(s_t+\mu(v) s_x=-{1\over 2\varepsilon} (r+s-F(r-s))\), with the transformed initial conditions. With this change the authors prove the existence of global smooth solutions to the original system.
scientific article; zbMATH DE number 1437330