Does quenching for degenerate parabolic equations occur at the boundaries? (Q2724368)

The authors treat the parabolic problem for \(u(x,t)\), NEWLINE\[NEWLINELu=x^q u_t-u_{xx}= f(u)\quad\text{in}\quad\Omega= D\times I_T,\quad u(x,0)= 0\quad\text{in}\quad D,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_x+ \alpha_1u= 0\quad\text{on}\quad\Gamma_1= \{0\}\times I_T\quad\text{and}\quad u_x+ \alpha_1 u=0\quad\text{on}\quad\Gamma_2=\{a\} \times I_T,\tag{2}NEWLINE\]NEWLINE where \(D=(0,a)\) and \(I_T=(0,T) \), \(T\leq\infty\), \(q>0\). Here \(\alpha_1= \alpha_2=0\) represents the second and \(\alpha_1< 0<\alpha_2\) the third boundary condition (BC). The function \(f(u)\) is convex, \(f>0\), \(f'>0\) and \(f(u)\to \infty\) as \(u\to c^-\), where \(c>0\) is given. NEWLINENEWLINENEWLINEQuenching means that the solutions exists in \(\Omega\) with \(T<\infty\) and \(\max_u(x,t)\to c\) as \(t\to T\). The main results are: In both cases there is only one quenching point. It is \(x=0\), with 2nd BC and a point \(x_0\in D\) with 3rd BC.