Multidimensional Goursat problem for semilinear hyperbolic equations
We are concerned with the following fully semilinear Goursat problem \[ Pu= f(x, t, u, \nabla u),\quad (x, t)\in \Gamma,\quad u|_{\partial \Gamma}= 0, \] where \((x, t)\in \mathbb{R}^3\), \(P\equiv P(x, t, \partial_{x_1}, \partial_{x_2}, \partial_t)= \partial^2_t+ \sum^2_{|\alpha|= 1} a_\alpha(x, t) \partial^\alpha \partial^{2- |\alpha|}+ R_1\). We have \(a_\alpha\in C^\infty_b(\Omega)\), where \(\Omega\) is a domain of \(\mathbb{R}^3_+\), and \(\mathbb{R}^3_+= \{(x, t)\mid t\geq 0, x\in \mathbb{R}^2\}\), \(R_1\) is an operator of order \(\leq 1\), \(f\) is a \(C^\infty\) function of its arguments, \(P\) is a strictly hyperbolic operator with respect to \(t\) in \(\mathbb{R}^3_+\), and \(\Gamma\) is the solid forward characteristic cone, with the vertex at the origin, of \(P\). Since this is a characteristic boundary problem, and the characteristic cone \(\Gamma\) has a singularity at the origin, away from the vertex the cone is smooth. Therefore, the regularity of the solution usually is very low and is very difficult to be improved. In this paper, we show that the solution is Lipschitz and is smooth away from the boundary. In comparison with the flat case, the main difficulties to this problem lie in the more complicated geometric nature, in the more technical commutation relation between \(P\) and the tangent boundary vector fields. In fact, we have to deal with curved characteristics. Another difficulty comes from the nonlinear term \(f\) which depends on \(u\) and \(\partial u\). Our result is partial and local, but for convenience, we can consider this problem globally for \(x\in \mathbb{R}^3\) by using the property of finite dependence domain for \(P\). We shall always suppose that the nonlinear term \(f(x, t, u, \partial u)\) is zero when \(t+ |x|\) is sufficiently large, and set \(\Omega_T= \mathbb{R}^3_+\cap \{t< T\}\) for any \(T> 0\).
scientific article; zbMATH DE number 913293
scientific article