Cauchy problem for a class of degenerate parabolic systems
A system of partial differential equations of the form \[ \partial u/\partial t = \sum_{\tilde k+2\tilde j\leq 1} A_{kj}(t,x) D^ k_{x'} B^ j_{x_ m}u = f\quad (t,x) \] where \(B_{x_ n}=\partial^ 2/\partial x^ 2_ n+((2\nu +1)/x_ n)\partial /\partial x_ n\) is the Bessel operator, \(\nu_ 0=2\nu +1>0\), \(\tilde k+2\tilde j=k_ 1/2b_ 1+...+k_{n-1}/2b_{n- 1}+2j/2b_ m,b_ 1\geq...\geq b_{n-1}\geq b_ n\) are integers is called \(2\vec B\)-parabolic in the domain \(\pi^+=(0,T)\times E^+_ n,E^+_ n=E_{n-1}\times (0,\infty)\) if the solution of the characteristic equation \[ \det \| \sum_{\tilde k+2\tilde j=1} A_{kj}(t,x)(i\delta ')^ k(-\delta^ 2_ n)^ j-\lambda E\| = 0 \] satisfies the inequality Re \(\lambda\) (t,x;\(\delta\) \({}_ 1,...,\delta_ n)\leq -\delta (\delta_ 1^{2b_ 1}+...+\delta_ n^{2b_ n})\), \(\delta =const>0\). Under certain conditions the unique solution of this Cauchy problem admits a Stieltjes integral representation with respect to a Borel measure. Estimation for the solution and for the Green matrix are given.
scientific article; zbMATH DE number 3888198