Global Cauchy problems modulo flat functions (Q1058665)

Let E be a real Banach space, \(\Omega\) an open subset of \(R^ n\), T a real number, \(0<T\leq \infty\), and f(x,t,u,p) a function of class \(C^{\infty}\) defined on \(\Omega \times [0,T)\times E\times L(R^ n,E).\) We consider the nonlinear Cauchy problem \((1)\quad u_ t=f(x,t,u,u_ x),\quad u(x,0)=u_ 0(x),\) with smooth initial data. In general problem (1) does not have exact solutions. In this paper necessary and sufficient conditions for the existence of a global ''\(\rho\)-flat solution'' of (1) are given. A function \(f\in C^{\infty}(X;E)\) is k-\(\rho\)-flat if for any integer \(m\leq k\) and any smooth differential operator P of order m in X, \(\rho^{-k+m}Pf\) is continuous. If f is k-\(\rho\)-flat for all k, we say that f is \(\rho\)-flat. Let be \(\phi =\phi (x,t)=\int^{t}_{0}\| f_ p(x,s,v(x,s),q(x,t))\| ds.\) A function \(u(x,t)\in C^{\infty}(M:E)\) is a \(\rho\)-flat solution of (1) if \(u_ t-f(x,t,u,u_ x)\) is \(\rho\)- flat, where M is an open subset of \(M_ 0\). When \(M=M_ 0\) we say the \(\rho\)-flat solution is global. The main theorem 1.4, p. 243 includes the following simple corollary: Assume (1) is quasilinear, i.e., \(f(x,t,u,p)=g(x,t,u)p.\) Then (1) has global \(\rho\)-flat solutions for every initial data. The theorem is illustrated by an example of the operator: \(L=\partial /\partial t- ib(x,t)\partial /\partial x+c(x,t).\)