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Nonlinear system of equations of mixed type and nonlinear complex equation of mixed type. (English) Zbl 0884.35106

Ding, Xiaxi (ed.) et al., Nonlinear evolutionary partial differential equations. Proceedings of the international conference, Beijing, China, June 21–25, 1993. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 3, 393-401 (1997).
In \(D\subset \mathbb{R}^2\) the nonlinear system \[ A_{11} u_{xx} -2A_{12} u_{xy} + A_{22} u_{yy} +B_1u_x +B_2u_y +Cu- \text{grad} F(u)=g \tag{*} \] is considered, where \(u(x,y)= (u_1(x,y), \dots, u_N(x,y))^T\), \(g(x,y)= (g_1(x,y), \dots, g_N(x,y))^T\) and \(A_{ii}= a^{ij} (x,y)I\;(i,j= 1,2)\), \(I\) the \(N\times N\) unit matrix, \(B_j= (b^j_{kl} (x,y))\), \(C= (c_{kl} (x,y))\) \((k,l=1, \dots, N)\) are \(N\times N\) symmetric matrices, and \[ (a^{12})^2 -a^{11} a^{22} \begin{cases} <0, \quad & y> 0\\ =0, \quad & y=0 \\ >0, \quad & y<0. \end{cases} \] Under conditions on \(a^{ij}\) and the nonlinear function \(F(u)\) the author proves the existence of a strong solution for the modified Tricomi problem. Beyond it a nonlinear complex equation of mixed type is considered which, by separating it into the real and imaginary part, can be written as (*).
For the entire collection see [Zbl 0869.00036].

MSC:

35M10 PDEs of mixed type
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