Existence and Nonexistence of Global Solutions of the Convective Porous Medium Equations with a Nonlinear Forcing at the Boundary

Abstract

This article deals with the existence and nonexistence of global positive solutions of the following equations: \\begin{array}{l@{\\hskip2em}l}u_t=(u^m)_{xx}+{\\displaystyle{\\mu\\over n}}(u^n)_x,&x\\in(0,1),\\ t>0 \\\\(u^m)_x|_{x=0}=0,&t>0 \\\\(u^m)_x|_{x=1}=au^p,&t>0 \\\\u(x,0)=u_o(x)>0.&x\\in[0,1].\\end{array} where m, n, μ, p, a>0. The necessary and sufficient conditions for the global existence of solutions are obtained.