Global solvability of nonlinear diffusion equations with forcing at the boundary
This paper deals with nonnegative solutions of the model problems \[ u^m_{xx} +{\varepsilon \over n} u^n_x=u_t\text{ for }0<x<1, t>0;\;u(x,0) =u_0(x)\text{ for }0\leq x\leq 1 \] and the boundary conditions \[ u(0,t)=0,\;u^m_x (1,t)=au^p(1,t)\text{ resp. }-u^m_x (0,t)= au^p(0,t),\;u(1,t) =0\text{ for }t>0. \] Here \(a,\varepsilon,m,n,p>0\), \(u_0\in L^\infty (0,1), u_0\geq 0\). By means of the limit solutions constructed in a previous paper [Comm. Partial Differential Equations 16, No. 1, 105-143 (1991; Zbl 0738.35033)] and of differential inequalities involving the \(L^{r+1}\) norm of solutions with \(r>m\) and \(r+1\geq \max(4n,2p)\), the author shows that, under rather general conditions on the data, solutions exist for all time. Possible generalizations to a more general class of problems are discussed in the final section of the paper.
scientific article; zbMATH DE number 1005693
scientific article