Semilinear anisotropic evolution partial differential equations. (Q1404888)

Consider the anisotropic Cauchy problem \(\partial_t u+P(D)u-G(\partial^\alpha u)=0\) for \(t>0\), \(u(x,0)=u_0(x)\), where \(P(D)=\sum_{| \alpha:\rho| \leq m} c_\alpha D_x^\alpha\) is a quasi-elliptic operator of (anisotropic) order \(m\) and \(G\) is a homogeneous polynomial with constant coefficients. Here \(\alpha\) is a multiindex, \(\rho\in(0,1]^n\), \(| \alpha:\rho| =\sum_{j=1}^n \alpha_j/\rho_j\). Using the fixed point technique the authors prove that under suitable assumptions on \(P\) and \(G\) the problem has a weak solution defined in a finite time interval. The global existence is proved by the authors in [Nonlinear Anal., Theory Methods Appl. 47, No. 1, 525--530 (2001; Zbl 1042.35562)] under more restrictive assumptions.