Viscous approach of discontinuous solutions of semilinear symmetric hyperbolic systems (Q818305)

Let \(A_i(t,x)\), \(0\leq i\leq n\), \(B(t,x), E_{il}(t,x)\), \(1\leq i\), \(l\leq n,(t,x)\in \mathbb{R}\times \mathbb{R}^n,\) be \(N\times N\) symmetric matrices whose entries are \(C^{\infty}(\mathbb R^{n+1})\) and constant outside a compact set. Let \(f\) and \( F\) be \(\mathbb R^N\)-valued, sufficiently smooth functions with \( F(t,x,0)=0.\) This paper deals with solutions \(u(t,x)\) of the hyperbolic system \[ P_Tu=A_0(t,x)\partial_tu+\sum_{1\leq i\leq n} A_i(t,x)\partial_iu+B(t,x)u-F(t,x,u)-f(t,x)=0 \tag{1} \] on \( \Omega_T=(0,T)\times\mathbb R^n\), which are regular outside a smooth hypersurface \(\Gamma\); \(\Gamma\) is supposed to be characteristic and of constant multiplicity. The allowed singularities are jumps of \(u\) or of its derivatives on \(\Gamma\). Without loss of generality \(\Gamma\) is assumed to be \(\{x_n=0\}\). Set \(\Gamma_T=\{(t,x)\in \mathbb R^{n+1}\), \(0\leq t\leq T\), \(x_n=0\}.\) We denote by \(p-H^{\infty}(\Omega_T)\) the space of functions whose all derivatives are \(L^2\) on both sides of \(\Gamma_T\). Consider the perturbed problem \[ P_T^{\varepsilon}=P_Tu-\varepsilon \mathcal Eu=0\quad\text{on}\;\Omega_T \] where \(\mathcal E=\sum_{1\leq i,l\leq n}\partial_iE_{il}(t,x)\partial_l \) is a suitable viscosity. Given \(T_0\) and a \(p-H^{\infty}(\Omega_{T_0})\) solution \(u_0\) of (1), the author shows the existence of \(T_1\in ]0,T_0]\), \(\varepsilon_0>0,\) and a family \(\{u_{\epsilon}\), \(0<\varepsilon\leq\varepsilon_0\}\) of solutions of \(P^{\varepsilon }_{T_1}(u_{\varepsilon})=0\) such that \(u_{\varepsilon}\to u_0\) in \(L^2(\Omega_{T_1})\), as \(\varepsilon\to 0.\) Asymptotic expansions of the form \[ u_{\varepsilon}(t,x)=\sum_{0\leq j\leq k}\sqrt{\varepsilon}^jU^j \biggl(t,x,\frac{x_n}{\varepsilon}, \frac{x_n}{\sqrt{\varepsilon}}\biggr)+ \sqrt{\varepsilon}^{k+1}r_{\varepsilon}(t,x) \] are obtained where \( U^0(t,x,z,\theta)\to u_0(t,x)\), as \(| z| ,| \theta| \to + \infty\). Related results were obtained by \textit{O. Guès} [Ann. Inst. Fourier 45, No. 4, 973--1006 (1995; Zbl 0831.34023)].