Blow-up solutions of degenerate parabolic problems

Abstract

In this article, we study the degenerate parabolic problem, x qut-(x βu x) x = x qf(u), satisfying the Dirichlet boundary condition and a nonnegative initial condition where q and β are given constants and f is a suitable function. We show that under certain conditions the degenerate parabolic problem has a blow-up solution and the blow-up set of such a blow-up solution is the whole domain of x. Furthermore, we give the sufficient condition to blow-up in finite time. Finally, we generalize the degenerate parabolic problem into the general form, k(x)u t-(p(x)u x) x =k(x)f(u). Under appropriate assumptions on functions k, p and f, we still obtain the same results as the previous problem.