documentclass[12pt,reqno]{amsart}
%------------------------------------------------------------
  usepackage{etoolbox}
   makeatletter
   patchcmdmaketitle
      {uppercasenonmathshorttitle}
        {}
        {}{}
  patchcmdmaketitle
        {@nxMakeUppercase{thetoks@}}
       {thetoks@}
         {}
         {}{}
    %----
   %patchcmd@settitle
   %  {uppercasenonmath@title}
   % {}
   %  {}{}
    patchcmd@setauthors
          {MakeUppercase{authors}}
          {authors}
           {}{}
 makeatother
  %------------------------------------------------------------
 %usepackage[notref,notcite]{showkeys}
  usepackage{amsmath,amssymb,amsthm}
  usepackage{color}
  usepackage{url}
   %-----------------
  %------------------------------------------------------------
  usepackage[utf8]{inputenc}
  usepackage[T1]{fontenc}
   %------------------------------------------------------------
 usepackage{geometry}
   geometry{left=2cm,right=2cm,top=2cm,bottom=2cm}

%-------------------------------------------
 address{$^{[1]}$ The address of author1.}
  email{url{Author1@....com}}
    address{$^{[2]}$ The address of author2.}
 email{url{Author2@....com}}

 subjclass[2010]{......}

    keywords{......}

  %%%%%%%%%%%% begin{document} %%%%%%%%%%%%%%%%%%%%
 begin{document}
 author[Author1, Author2] {Author1$^{1}$ and Author2$^{2}$ }
 title{Title}
 maketitle
 end{document}