Multidimensional Goursat problem for semilinear hyperbolic equations
type: Item
instance of: scholarly article
title: Multidimensional Goursat problem for semilinear hyperbolic equations
zbMATH Open document ID: 0854.35072
DOI: 10.1007/BF02662179
author: Daoyuan Fang
author: Xiaoqing Gong
published in: Applied Mathematics. Series B (English Edition)
publication date: 1997-01-20T00:00:00Z
review text: 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 \(T> 0\).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 913293
zbMATH Keywords: semilinear Goursat problem
zbMATH Keywords: characteristic boundary problem
MaRDI profile type: Publication
cites work: The Cauchy problem for semilinear hyperbolic equations with cross discontinuous data
cites work: Multidimensional Goursat problem for semilinear wave equations
cites work: Blow-up for solutions of □u=|u|pwith small initial data.
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description: scientific article; zbMATH DE number 913293