Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

We study the Cauchy problem in $\re \times \re_+$ for one-dimensional 2mth-order, m>1, semilinear parabolic PDEs of the form ($D_x=\partial/\partial x$) \[ u_t = \Dx u + |u|^{p-1}u, \,\,\,\mbox{where} \,\, \,\,\,p > 1, \quad \mbox{ and } \quad u_t = \Dx u + e^u \] with bounded initial data u0 (x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical second-order parabolic equations ut = uxx + up and ut = uxx + eu from combustion theory, the blow-up in their higher-order counterparts is asymptotically self-similar. In particular, there exist exact nontrivial self-similar blow-up solutions, u* (x,t) = (T-t)-1/(p-1)f (y) in the case of the polynomial nonlinearity and u(x,t) = -ln(T-t) + f(y) for the exponential nonlinearity, where y= x/(T-t)1/2m is the backward higher-order heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non--self-adjoint higher-order linear dif...