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

Abstract

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 differential operators. We show that there are at least $2 \lfloor \frac m2 \rfloor$ nontrivial self-similar solutions to the full PDEs. Numerical solution of the ODEs for m=2 and 3 supports this, and the time dependent solutions of the PDEs for m=2 are then studied by using a scale invariant adaptive numerical method. It is shown that those functions f(y), which have the simplest spatial shape (e.g., a single maximum), correspond to stable self-similar solutions. A further countable subset of nonsimilarity blow-up patterns can be constructed by linearization and matching with similarity solutions of a first-order Hamilton--Jacobi equation.