Mixed problems for evolution equations
Let A(t,x;\(\tau\),\(\xi)\) and \(B_ j(t,x;\tau,\xi)\), \(j=1,2,...,m\), be polynomials with respect to (\(\tau\),\(\xi)\) with coefficients depending on the parameters (t,x). Let us consider a mixed problem (P): \[ A(t,x;D_ t,D_ x)u=f\quad in\quad (0,T)\times {\mathbb{R}}^ n_+, \] \[ B_ j(t,x;D_ t,D_ x)u=g_ j\quad on\quad (0,T)\times \partial {\mathbb{R}}^ n_+,\quad j=1,2,...,m, \] \[ D^ k_ tu=u_ k\quad on\quad \{0\}\times {\mathbb{R}}^ n_+,\quad k=1,2,...,\mu -1, \] where f, \(g_ j\), \(u_ i\) are given. The conditions in terms of the Newton polygon of A are investigated, under which the problem (P) is \(H^{\infty}\)-well posed.
scientific article; zbMATH DE number 3912872